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8c3b5f62548d3f54793c560e639e8d173c7ef50d | subsection | 78 | 174 | Substitutions | Therefore \left(\!\begin{}(x,t)\\ \cdot \end{}\!\right)_{(x,t)\in \coprod X}(P) is a mapping from \prod _{(x,t)\in \coprod X}\mathrm {T}_{\Sigma }(X)_{t}^{\vert P \vert _{x}} to \mathrm {T}_{\Sigma }(X)_{s}.Let \mathcal {T} be the subset of \mathrm {T}_{\Sigma }(X) defined, for every s\in S, as follows:\mathcal {T}_{s}... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.0... |
6a285ac02fd14cbd9d714d7253d51018116200b1 | subsection | 79 | 174 | Substitutions | Then, for every t\in S and every x\in X_{t}, we have that\textstyle \vert \sigma ((P_{i})_{i\in \vert w \vert }) \vert _{x}=\sum _{i\in \vert w \vert }\vert P_{i} \vert _{x}.But we have that\left(\!\begin{}(x,t)\\(Q^{x,t}_{\alpha })_{\alpha \in \vert \sigma ((P_{i})_{i\in \vert w \vert }) \vert _{x}}\end{}\!\right)_{(x... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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... |
e46a035146f2ed92afc1eb84cc9b0c7a89121dcf | subsection | 80 | 174 | Substitutions | Therefore \left(\!\begin{}z\\ \cdot \end{}\!\right)(P) is a mapping from \mathrm {T}_{\Sigma }(X)^{\vert P \vert _{z}}_{u} to \mathrm {T}_{\Sigma }(X)_{s}.Remark For every S-sorted set X the family \left(\left(\!\begin{}(x,t)\\ \cdot \end{}\!\right)_{(x,t)\in \coprod X}(P)\right)_{s\in S} of global substitution operato... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.01407053042203188... |
f4a64081a50cdd67c405bad2a805732592ad1f92 | subsection | 81 | 174 | Substitutions | But for W, either (1) W=z, for a unique z\in X_{s} , or (2) W=\sigma , for a unique \sigma \in \Sigma _{\lambda , s}, or (3) W=\sigma ((W_{i})_{i\in \vert w \vert }), for a unique w\in S^{\star }-\lbrace \lambda \rbrace , a unique \sigma \in \Sigma _{w,s}, and a unique family (W_{i})_{i\in \vert w \vert }\in \mathrm {T... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.0002878344675991684,
-0.019... |
141e28881d420bbaba9005c84584e589d65294b6 | subsection | 82 | 174 | Substitutions | To prove the statement, we need to show that g_{s}(\sigma )=g_{s}(\sigma ), which trivially holds.Finally, in case (3), we have the following equationsg_{s}\left(\left(\!\begin{}(x,t)\\(P^{x,t}_{\alpha })_{\alpha \in \vert W \vert _{x}}\end{}\!\right)_{(x,t)\in \coprod X}(W)\right)&=g_{s}\left(\left(\!\begin{}(x,t)\\(P... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.018277034163475037,
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0.002749945968389511,
... |
ad3a2db47db3c3ad01a58052e254361e5c440c0b | subsection | 83 | 174 | Substitutions | Then, for every \left((P^{x,t}_{\alpha })_{\alpha \in \vert W \vert _{x}}\right)_{(x,t)\in \coprod X}, \left((Q^{x,t}_{\alpha })_{\alpha \in \vert W \vert _{x}}\right)_{(x,t)\in \coprod X}\in \prod _{(x,t)\in \coprod X}\mathrm {T}_{\Sigma }(X)_{t}^{\vert W \vert _{x}}, if, for every (x,t)\in \coprod X and every \alpha ... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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97b1883f45cc01ae9582187a75889c9bc9f4739c | subsection | 84 | 174 | Substitutions | We remind the reader that in accordance with what is established in Proposition REF , we have let, for abbreviation, for every (w,s)\in S^{\star }\times S and every \sigma \in \Sigma _{w,s}, \sigma stand for F_{\sigma }^{\mathbf {T}_{\Sigma }(X)}, the structural operation of \mathbf {T}_{\Sigma }(X) associated to \sigm... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.... |
30cfe843d3f117e9c39798a92c12a5e9bdc78182 | subsection | 85 | 174 | Substitutions | Then, for the S-sorted mapping \left(\!\begin{}z\\L\end{}\!\right) from X to \mathrm {T}_{\Sigma }(X)^{\wp } defined as: \left(\!\begin{}z\\L\end{}\!\right)_{u} is the mapping from X_{u} to \mathrm {Sub}(\mathrm {T}_{\Sigma }(X)_{u}) that sends z to L and y\in X_{u}-\lbrace z\rbrace to \lbrace y\rbrace , while, for t\i... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.009300352074205875,
0.0002286463713971898,
0... |
3566473c7d514be32f1084dbcc94b9b98634cdb7 | subsection | 86 | 174 | Substitutions | Then we know that P either has the form (1) z, for a unique z\in X_{s}, or (2), \sigma , for a unique \sigma \in \Sigma _{\lambda , s}, or (3) \sigma ((P_{i})_{i\in \vert w \vert }), for a unique w\in S^{\star }-\lbrace \lambda \rbrace , a unique \sigma \in \Sigma _{w,s}, and a unique family (P_{i})_{i\in \vert w \vert... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.01277905609458685,
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-... |
5f0b8dd3e3e43e0d26f0a9b748bc505c183d6dc0 | subsection | 87 | 174 | Substitutions | Let us note that, since, for every x\in X_{t}, L_{x}^{\vert P \vert _{x}} is embedded into \mathrm {T}_{\Sigma }(X)_{t}^{\vert P \vert _{x}}, \prod _{x\in X_{t}}L_{x}^{\vert P \vert _{x}} is also embedded into \prod _{x\in X_{t}}\mathrm {T}_{\Sigma }(X)_{t}^{\vert P \vert _{x}}. Therefore \prod _{(x,t)\in \coprod X}L_{... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.005264685023576021,
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0.02029574289917946,
-0... |
e5b168648c981c29fce34f2f0f13d72b787be740 | subsection | 88 | 174 | Substitutions | Then, for every S-sorted mapping \left(\left(\!\begin{}x\\L_{x}\end{}\!\right)_{x\in X_{t}}\right)_{t\in S} from X to \mathrm {T}_{\Sigma }(X)^{\wp }, we have that\left(\left(\left(\!\begin{}x\\L_{x}\end{}\!\right)_{x\in X_{t}}\right)_{t\in S}\right)^{\sharp }_{s}(P) = \mathrm {Im}\left(\left(\!\begin{}(x,t)\\ \cdot \e... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
-0.029415642842650414,
-0.0015371503541246057,
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0.015081619843840599,
0.006545285694301128,
0.03429790586233139,
0.00197197706438601,
0.006980112288147211,
0... |
6e3396d9a8bbe72c87cefc32dd8a6249c23c30a9 | subsection | 89 | 174 | Substitutions | Therefore, since, for every Q\in L_{z}, it happens that \left(\!\begin{}z\\Q\end{}\!\right)(z) = Q, we have that \mathrm {Im}\left(\left(\!\begin{}(x,t)\\ \cdot \end{}\!\right)_{(x,t)\in \coprod X}(P)\right) = L_{z}. Thus \mathrm {Im}\left(\left(\!\begin{}(x,t)\\ \cdot \end{}\!\right)_{(x,t)\in \coprod X}(P){\textstyle... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
-0.031926315277814865,
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0.014032622799277306,
0.026127079501748085,
0.010133400559425354,
0.0023521240800619125,
0.0013859027530997... |
3f870eeb0f7acdd312fc8b5d194fee72e581e13c | subsection | 90 | 174 | Substitutions | Consequently\mathrm {Im}\left(\left(\!\begin{}(x,t)\\ \cdot \end{}\!\right)_{(x,t)\in \coprod X}(P){\textstyle \upharpoonright }_{\prod _{(x,t)\in \coprod X}L_{x}^{\vert P \vert _{x}}}\right) = \lbrace \sigma \rbrace .Let (w,s)\in (S^{\star }-\lbrace \lambda \rbrace )\times S, \sigma \in \Sigma _{w,s}, and let (P_{i})_... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
-0.03827444463968277,
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... |
592d722fdfefcbc963d3789dd78cf39b4d75ffe0 | subsection | 91 | 174 | Substitutions | Then, since \left(\left(\left(\!\begin{}x\\L_{x}\end{}\!\right)_{x\in X_{t}}\right)_{t\in S}\right)^{\sharp } is a homomorphism from \mathbf {T}_{\Sigma }(X) to \mathbf {T}_{\Sigma }(X)^{\wp }, we have that\left(\left(\left(\!\begin{}x\\L_{x}\end{}\!\right)_{x\in X_{t}}\right)_{t\in S}\right)^{\sharp }_{s}(\sigma ((P_{... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.006365586072206497,
-0.004573000594973... |
019e8698e9676e9b07b197ac59c3e415a3cd9cd2 | subsection | 92 | 174 | Substitutions | Hereby completing our proof.Remark Given an S-sorted mapping \left(\left(\!\begin{}x\\L_{x}\end{}\!\right)_{x\in X_{t}}\right)_{t\in S} from X to \mathrm {T}_{\Sigma }(X)^{\wp }, and P a term in \mathrm {T}_{\Sigma }(X)_{s}, for some s\in S, we have that
\textstyle \left(\left(\left(\!\begin{}x\\L_{x}\end{}\!\right)_{... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
-0.03811284154653549,
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0.010939515195786953,
-0.009635012596845627,
-... |
58d37bf8d9bb3112641598adf9b704d50d43774e | subsection | 93 | 174 | Substitutions | Then\textstyle \left(\!\begin{}z\\L\end{}\!\right)^{\sharp }_{s}(P) = \mathrm {Im}\left(\left(\!\begin{}z\\ \cdot \end{}\!\right)(P){\textstyle \upharpoonright }_{L^{\vert P \vert _{z}}}\right).Definition 3.12 Let \left(\left(\!\begin{}x\\L_{x}\end{}\!\right)_{x\in X_{t}}\right)_{t\in S} be an S-sorted mapping from X t... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
-0.04256010800600052,
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0.013568894937634468,
0.009457802399992943,
0.04173636436462402,
0.023766541853547096,
-0.004153043031692505,... |
a340060c47d1bf17650cd84f32b80e8f395e598d | subsection | 94 | 174 | Substitutions | Then we will denote by \left(\!\begin{}z\\L\end{}\!\right)^{\sharp \mathfrak {p}} the canonical extension of the underlying mapping of the homomorphism \left(\!\begin{}z\\L\end{}\!\right)^{\sharp } from \mathbf {T}_{\Sigma }(X) to \mathbf {T}_{\Sigma }(X)^{\wp }.Remark Let \left(\left(\!\begin{}x\\L_{x}\end{}\!\right)_... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1386,
"openalex_id": "https://openalex.org/W1593799327",
"raw": "F. Gécseg and M. Steinby, Tree automata. Akadémiai Kiadó, Budapest, 1984.",
"source_ref_id": "10d97f7641a8e8d5461b308c0a65424c3049f438",
"start": 1318
... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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625e48f7e1ce212b1472f5fc0e2e7bf16de5c8e0 | subsection | 95 | 174 | Substitutions | The proposition states that, for every sort s\in S, every s-recognizable language K, and every operator \left(\left(\!\begin{}x\\L_{x}\end{}\!\right)_{x\in X_{t}}\right)_{t\in S} such that, for every t\in S and every x\in X_{t}, L_{x}\in \mathrm {Rec}_{t}(\mathbf {T}_{\Sigma }(X)),
\left(\left(\left(\!\begin{}x\\L_{x}\... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.007467115763574839,
0.0... |
8bfbbdb6d17016dd07418c8e8da0e505b7f10beb | subsection | 96 | 174 | Substitutions | Moreover, k and \mathcal {W}_{\Phi } denote the S-sorted sets (k_{r})_{r\in S} and (\mathcal {W}_{\Phi _{r}})_{r\in S}, respectively.Let \Psi =(\Psi _{r})_{r\in S} be the binary relation on \mathrm {T}_{\Sigma }(X) defined as follows: For every r\in S, \Psi _{r} is the subset of \mathrm {T}_{\Sigma }(X)_{r}^{2} consist... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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f5a8fd5adcda525855e0576263c97a1bd01a837b | subsection | 97 | 174 | Substitutions | We want to show that\left(\sigma ((P_{i})_{i\in \vert w \vert }), \sigma ((Q_{i})_{i\in \vert w \vert })\right)\in \Psi _{u}.Let us note that, by definition of \Psi , for every i\in \vert w \vert , we have that (P_{i}, Q_{i})\in \Phi _{w_{i}}. Since \Phi is a congruence on \mathrm {T}_{\Sigma }(X), we conclude that \le... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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98ee98ecd37fd716adba3ed8533ab919ebac438f | subsection | 98 | 174 | Substitutions | It follows that, for every i\in \vert w \vert , for every t\in S and x\in X_{t}, \vert P_{i} \vert _{x}=0. Therefore, for every i\in \vert w \vert , the term P_{i} is a term in \left(\left(\left(\!\begin{}x\\L_{x}\end{}\!\right)_{x\in X_{t}}\right)_{t\in S}\right)^{\sharp \mathfrak {p}}_{w_{i}}([P_{i}]_{\Phi _{w_{i}}})... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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485b2824236df4bf312d9c64efa34d9bb61dda7a | subsection | 99 | 174 | Substitutions | Let \left((V^{x,t}_{\beta })_{\beta \in \vert W^{\ddagger } \vert _{x}}\right)_{(x,t)\in \coprod X} be the element in \prod _{(x,t)\in \coprod X}L_{x}^{\vert W^{\ddagger } \vert _{x}} obtained by joining, in order, the family \left(\left((V^{x,t,i}_{\beta })_{\beta \in \vert W^{\ddagger _{i}} \vert _{x}}\right)_{(x,t)\... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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ccba4bf9df0264d81f11181490665593ce14fa9d | subsection | 100 | 174 | Substitutions | Therefore, from Equation REF we obtain the following equation\left(\!\begin{}(x,t)\\(U^{x,t}_{\alpha })_{\alpha \in \vert W^{\dagger } \vert _{x}}\end{}\!\right)_{(x,t)\in \coprod X}(W^{\dagger })=
\left(\!\begin{}(x,t)\\(U^{x,t}_{\alpha })_{\alpha \in \vert W^{\dagger } \vert _{x}}\end{}\!\right)_{(x,t)\in \coprod X}(... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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e419ec3634df2774940ca4c8e39a44f05e0be134 | subsection | 101 | 174 | Substitutions | From Equation REF , for every i\in \vert w \vert ,P_{i}=\left(\left(\!\begin{}(x,t)\\(U^{x,t}_{\alpha +\sum _{j\in i}\vert W^{\dagger _{j}} \vert _{x}})_{\alpha \in \vert W^{\dagger _{i}} \vert _{x}}\end{}\!\right)_{(x,t)\in \coprod X}(W^{\dagger _{i}})\right).Hence, for every i\in \vert w \vert ,
P_{i}\in \left(\left(... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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2a7c96730b5132cdfa62d21e960d61efbf180138 | subsection | 102 | 174 | Substitutions | Therefore, for every i\in \vert w \vert , there are W^{\ddagger _{i}}\in [W^{\dagger _{i}}]_{\Phi _{w_{i}}} and \left((V^{x,t,i}_{\beta })_{\beta \in \vert W^{\ddagger _{i}} \vert _{x}}\right)_{(x,t)\in \coprod X} in \prod _{(x,t)\in \coprod X}L_{x}^{\vert W^{\ddagger _{i}} \vert _{x}} such thatQ_{i}=\left(\!\begin{}(x... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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c2ec13817f9cc7f8d75a630f62102a0a9a9b6cdd | subsection | 103 | 174 | Substitutions | Note that, by definition of \Psi , for every r\in S and every l\in k_{r}, \left(\left(\left(\!\begin{}x\\L_{x}\end{}\!\right)_{x\in X_{t}}\right)_{t\in S}\right)^{\sharp \mathfrak {p}}_{r}([W_{r,l}]_{\Phi _{r}}) is \Psi _{r}-saturated. Moreover,\textstyle \left(\left(\left(\!\begin{}x\\L_{x}\end{}\!\right)_{x\in X_{t}}... | {
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... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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71e01bb768497bf395fab1744cfea9941bbf679c | subsection | 104 | 174 | Substitutions | Then the language \sigma ^{\wp }((L_{i})_{i\in \vert w \vert })\in \mathrm {Rec}_{s}(\mathbf {T}_{\Sigma }(X)).It follows from Proposition REF and Proposition REF .Corollary 3.16 The S-sorted set (\mathrm {Rec}_{s}(\mathbf {T}_{\Sigma }(X)))_{s\in S} is a subalgebra of \mathbf {T}_{\Sigma }(X)^{\wp }. We will denote by... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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6c24fcf3163c249f6a6b1a04592458b27a41ec44 | subsection | 105 | 174 | Iterations | In this subsection we introduce the notion of iteration of a language with respect to a variable with the aim of proving that, when the considered language is recognizable, then its iteration with respect to a variable is also a recognizable language.We begin by stating the many-sorted counterpart of the single-sorted ... | {
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{
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"raw": "F. Gécseg and M. Steinby, Tree automata. Akadémiai Kiadó, Budapest, 1984.",
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"start": 251
... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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22bce774cd8d26de2e0ea6be3f35caa48f0e8f30 | subsection | 106 | 174 | Iterations | If L\in \mathrm {Rec}_{s}(\mathbf {T}_{\Sigma }(X)), then L^{\star \,z}\in \mathrm {Rec}_{s}(\mathbf {T}_{\Sigma }(X)).Let \Phi be the congruence on \mathbf {T}_{\Sigma }(X) defined as follows:\Phi = \Omega ^{\mathbf {T}_{\Sigma }(X)}(\delta ^{s,L})\cap \Omega ^{\mathbf {T}_{\Sigma }(X)}(\delta ^{s,z}).By Proposition R... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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c9bce550a32aa62a2839d00231a323a8e53ad47c | subsection | 107 | 174 | Iterations | Moreover, for every r\in S, the index of \Psi _{r} on \mathrm {T}_{\Sigma }(X)_{r} is bounded by k_{r}2^{k_{r}}. Consequently, the S-sorted \mathrm {T}_{\Sigma }(X)/{\Psi } is finite.Let us check that \Psi is a congruence on \mathbf {T}_{\Sigma }(X). Let (w,u)\in (S^{\star }-\lbrace \lambda \rbrace \times S), \sigma \i... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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1aff8ab8d6eb03bc2359928b44eae55366d4332b | subsection | 108 | 174 | Iterations | Assume that\sigma ((P_{i})_{i\in \vert w \vert }))\in \left(\!\begin{}z\\L^{\star \, z}\end{}\!\right)^{\sharp \mathfrak {p}}_{u}([W_{u,l}]_{\Phi _{u}}).Then there are W^{\dagger }\in [W_{u,l}]_{\Phi _{s}} and (U^{z}_{\alpha })_{\alpha \in \vert W^{\dagger } \vert _{z}} in (L^{\star z})^{\vert W^{\dagger } \vert _{z}} ... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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6838aa6f48f76c319f8851069864c8968b6c5af7 | subsection | 109 | 174 | Iterations | Therefore, we have that\sigma ((P_{i})_{i\in \vert w \vert })\in L^{j,z}=L^{j-1,z}\cup \left(\!\begin{}z\\L^{j-1,z}\end{}\!\right)^{\sharp \mathfrak {p}}_{s}(L).By the minimality of j, we conclude that \sigma ((P_{i})_{i\in \vert w \vert })\in \left(\!\begin{}z\\L^{j-1,z}\end{}\!\right)^{\sharp \mathfrak {p}}_{s}(L). T... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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27ba104fc129601b9c3ec32dc0c70501b8a981eb | subsection | 110 | 174 | Iterations | It follows that \sigma ((Q_{i})_{i\in \vert w \vert }) is a term in \left(\!\begin{}z\\L^{\star \, z}\end{}\!\right)^{\sharp \mathfrak {p}}_{s}([W_{s,l}]_{\Phi _{s}}), as desired.In case (b.1.ii), where we are assuming that \vert \overline{W}^{\dagger } \vert _{z}\ne 0, we claim that \overline{W}^{\dagger } cannot be t... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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e0b73fb7ed74b4a727b6b4252081d033253e9af8 | subsection | 111 | 174 | Iterations | Therefore, for every i\in \vert w \vert , there are \overline{W}^{\ddagger _{i}}\in [\overline{W}^{\dagger _{i}}]_{\Phi _{w_{i}}} and (\overline{V}^{z,i}_{\beta })_{\beta \in \vert \overline{W}^{\ddagger _{i}} \vert _{z}}\in (L^{\star \, z})^{\vert \overline{W}^{\ddagger _{i}} \vert _{z}} such thatQ_{i} = \left(\!\begi... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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1b2d34138c4ec6639713fba8bf7fc4ca5c875e20 | subsection | 112 | 174 | Iterations | On the whole, we conclude that\sigma ((Q_{i})_{i\in \vert w \vert })\in \left(\!\begin{}z\\L^{t, z}\end{}\!\right)^{\sharp \mathfrak {p}}_{s}(L)\subseteq L^{\star \, z}.Therefore \sigma ((Q_{i})_{i\in \vert w \vert }) is a term in \left(\!\begin{}z\\L^{\star \, z}\end{}\!\right)^{\sharp \mathfrak {p}}_{s}([W_{s,l}]_{\P... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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8495e8d5ab413f73d9aa93ba96d85017c40136ef | subsection | 113 | 174 | Quotients | We next define the notion of quotient of a language by another with respect to a variable of a specified sort with the aim of proving that, when one of the languages is recognizable, then the resulting quotient is also recognizable.We begin by stating the many-sorted counterpart of the single-sorted notion of z-quotien... | {
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... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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a911bfe0b5baccf7503630c22890d603222416bc | subsection | 114 | 174 | Quotients | Moreover, for the sort s\in S, \Phi _{s} saturates L.From now on, for every r\in S, let k_{r} and \mathcal {W}_{\Phi _{r}}=\lbrace W_{r,l}\mid l\in k_{r}\rbrace stand for the index of \Phi _{r} and a fixed transversal of \mathrm {T}_{\Sigma }(X)_{r}/{\Phi _{r}} in \mathrm {T}_{\Sigma }(X)_{r}, respectively. Moreover, k... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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371222f92e7f54d533d3dc809920c712d115533d | subsection | 115 | 174 | Quotients | Moreover\textstyle K^{-z}L=\bigcup _{W_{s,l}\in \mathcal {W}_{\Phi _{s}} [W_{s,l}]_{\Phi _{s}}\subseteq L}K^{-z}[W_{s,l}]_{\Phi _{s}}.Hence, K^{-z}L is \Psi _{s}-saturated because it is a finite union of \Psi _{s}-saturated languages. The statement follows from Proposition REF . | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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f4b831d31ccec0a48fe1c45d1b3c7ca783e12a21 | subsection | 116 | 174 | Tree Homomorphisms | Tree automata and tree homomorphisms were defined for the first time by Thatcher in . In the just cited paper Thatcher proved, among other things, that linear tree homomorphisms preserve recognizability. We shall now consider a class of many-sorted homomorphisms, the tree homomorphisms—which are the generalization to t... | {
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many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
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60dfa804984dd81ab9bc15615c55a58e188d897d | subsection | 117 | 174 | Tree Homomorphisms | Then, for a standard T-infinite countable T-sorted set of variables V^{T} = (\lbrace v^{t}_{n}\mid n\in \mathbb {N}\rbrace )_{t\in T}, which is assumed to be disjoint from all other alphabets, we will denote by V^{T}_{\downarrow \varphi ^{\star }(w)} = (V^{T}_{(\downarrow \varphi ^{\star }(w))_{t}})_{t\in T} the T-sort... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
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e783f3aa7d5b69def6472be578d3bee4df63216d | subsection | 118 | 174 | Tree Homomorphisms | Moreover, for every i\in \vert w \vert , the number of variables of type \varphi (w_{i}) is \vert \varphi ^{\star }(w) \vert _{\varphi (w_{i})} (while, for t\in T-\mathrm {Im}(\varphi ^{\star }(w)), V^{T}_{(\downarrow \varphi ^{\star }(w))_{t}} = \varnothing ).Remark For V^{T}_{\downarrow \varphi ^{\star }(w)}, if we d... | {
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} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
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77639b2236499e315475b3d084128e9493d0d75a | subsection | 119 | 174 | Tree Homomorphisms | Therefore, \downarrow \! v^{t}_{\vert \varphi ^{\star }(w) \vert _{t}} = \varnothing , if t\notin \mathrm {Im}(\varphi ^{\star }(w)); while \downarrow \! v^{t}_{\vert \varphi ^{\star }(w) \vert _{t}} = \lbrace v^{t}_{j}\mid j\in \vert \varphi ^{\star }(w) \vert _{t}\rbrace , if t\in \mathrm {Im}(\varphi ^{\star }(w)).T... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
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] | [
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44ee214ed4055c4cc364d8c1531dbab4d3651d0e | subsection | 120 | 174 | Tree Homomorphisms | We will say that a hyperderivor (\mathbf {c},f) from (\mathbf {\Sigma },X) to (\mathbf {\Xi },Y) is linear if, for every (w,s)\in S^{\star }\times S, every \sigma \in \Sigma _{w,s}, and every i\in \vert w \vert , no variable v^{\varphi (w_{i})}_{i} appears more than once in c_{w,s}(\sigma ), i.e., \vert c_{w,s}(\sigma ... | {
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many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
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01c7e3359c760690058c52d0ee0e558b6314fa80 | subsection | 121 | 174 | Tree Homomorphisms | On the basis of this isomorphism, we let \left(\!\begin{}v^{\varphi (w_{i})}_{i}\\ P_{i}\end{}\!\right)_{i\in \vert w \vert } stand for the T-sorted mapping from \downarrow \!\varphi ^{\star }(w) to \mathrm {T}_{\Xi }(Y) canonically associated to (P_{i})_{i\in \vert w \vert }\in \mathrm {T}_{\Xi }(Y)_{\varphi ^{\star }... | {
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} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
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e3262a0be4ef599dc4acd16175b1f3de576b83bd | subsection | 122 | 174 | Tree Homomorphisms | Then, finally, we let \mathrm {S}^{w}_{(P_{i})_{i\in \vert w \vert }} stand for \left[\mathrm {id}_{\mathbf {T}_{\Xi }(Y)},\left(\left(\!\begin{}v^{\varphi (w_{i})}_{i}\\ P_{i}\end{}\!\right)_{i\in \vert w \vert }\right)^{\sharp }\right], the unique homomorphism from \mathbf {T}_{\Xi }(Y\cup \downarrow \!\varphi ^{\sta... | {
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} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
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f06a6cfb6f8461bb604adb46883913cab12299db | subsection | 123 | 174 | Tree Homomorphisms | Then the S-sorted set \mathrm {T}_{\Xi }(Y)_{\varphi } is equipped, in a natural way, with a structure of \Sigma -algebra.Let \mathbf {c}(\mathbf {T}_{\Xi }(Y)) be the \Sigma -algebra defined as follows: The underlying S-sorted set of \mathbf {c}(\mathbf {T}_{\Xi }(Y)) is \mathrm {T}_{\Xi }(Y)_{\varphi } while, for (w,... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
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544a3e547028e53dfb4f05973a9c116d1086a08d | subsection | 124 | 174 | Tree Homomorphisms | Consequently, the operation \sigma ^{\mathbf {c}(\mathbf {T}_{\Xi }(Y))} is well-defined.Definition 3.25 Let (\mathbf {c},f) be a hyperderivor from (\mathbf {\Sigma },X) to (\mathbf {\Xi },Y). Then the unique homomorphism f^{\sharp } from \mathbf {T}_{\Sigma }(X) to \mathbf {c}(\mathbf {T}_{\Xi }(Y)) such that f^{\shar... | {
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... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
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a2be1d587c39abc7358698865884fcb192a3451c | subsection | 125 | 174 | Tree Homomorphisms | Then we denote by F_{\sigma }^{\mathbf {c}(\mathbf {A})} the mapping from A_{\varphi ^{\star }(w)} to A_{\varphi (s)} defined as follows:F_{\sigma }^{\mathbf {c}(\mathbf {A})}
\left\lbrace
\begin{array}{@{\:}c@{\:}c@{\:}l}
A_{\varphi ^{\star }(w)} &\usebox {
}& A_{\varphi (s)} \\
(a_{i})_{i\in \vert w \vert } &\longm... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
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305332caa7b549841a91c15ebaa49e5e63cbe65b | subsection | 126 | 174 | Tree Homomorphisms | In fact, let (a_{i})_{i\in \vert w \vert } be an element of A_{\varphi ^{\star }(w)} and (P_{i})_{i\in \vert w \vert }, (Q_{i})_{i\in \vert w \vert } elements of \mathrm {T}_{\Lambda }(Y)_{\varphi ^{\star }(w)} such that, for every i\in \vert w \vert ,g_{\varphi (w_{i})}(P_{i}) = a_{i} = g_{\varphi (w_{i})}(Q_{i}).But ... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
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55a88fcdf2750b18ad1d8a7fb7d9aa9f3b02ccce | subsection | 127 | 174 | Tree Homomorphisms | Thereforeg_{\varphi (s)}(\sigma ^{\mathbf {c}(\mathbf {T}_{\Xi }(Y))}((P_{i})_{i\in \vert w \vert })) = g_{\varphi (s)}(\sigma ^{\mathbf {c}(\mathbf {T}_{\Xi }(Y))}((Q_{i})_{i\in \vert w \vert })).It is obvious that g_{\varphi } is a homomorphism of \Sigma -algebras.By construction g^{\sharp }_{\varphi } is a homomorph... | {
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... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
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d075c5921fbcc348b376387c3d19f4b50be859d5 | subsection | 128 | 174 | Tree Homomorphisms | The proposition states that, for a sort s\in S, if the language L is s-recognizable, then its direct image by the s-th coordinate of a linear tree homomorphism f^{\sharp } is \varphi (s)-recognizable.Assumption To prove the following proposition we will assume that S, T, \Sigma and X are finite.Proposition 3.28
Let (\... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
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691951c44076e434e4226c0f59c46ce53a4db6bf | subsection | 129 | 174 | Tree Homomorphisms | Moreover, for s\in S, L is \Theta _{s}-saturated.From now on, for every r\in S, k_{r} and \mathcal {W}_{\Theta _{r}}=\lbrace W_{r,l}\mid l\in k_{r}\rbrace stand for the index of \Theta _{r} and a fixed transversal of \mathrm {T}_{\Sigma }(X)/{\Theta _{r}} in \mathrm {T}(X)_{r}, respectively.
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"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
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898e532a1e23d6b3bd519054aa2f77e73ed3c144 | subsection | 130 | 174 | Tree Homomorphisms | Moreover, if a=\operatorname{card}(\mathrm {T}_{\Xi }(Y)/{\Phi }), b=\operatorname{card}(\Sigma ),d = \max \lbrace \mathrm {card}(\mathrm {Subt}(c_{w,r}(\sigma ))_{t})\mid (w,r)\in S^{\star }\times S,\, \sigma \in \Sigma _{w,r},\, t\in T\rbraceand e=\max \lbrace \vert w \vert \mid (w,r)\in S^{\star }\times S \!\!\Sigma... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
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0a9cd515669a6392c2c2a01c34c1b84135a87b90 | subsection | 131 | 174 | Tree Homomorphisms | So the pair (\xi ((M_{j})_{j\in \vert u \vert }),\xi ((N_{j})_{j\in \vert u \vert })) satisfies the first condition for being related under \Psi _{t}.Regarding the second condition, let us assume that, for (w,r)\in S^{\star }\times S, \sigma \in \Sigma _{w,r}, R\in \mathrm {Subt}(c_{w,r}(\sigma ))_{t}, i\in \vert w \ve... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
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] | 2,018 | en | Computer Science | [
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832f1587cf729c94619926f00bf421f25d141287 | subsection | 132 | 174 | Tree Homomorphisms | Moreover, since R\in \mathrm {Subt}(c_{w,r}(\sigma ))_{t}, we have that, for every j\in \vert u \vert , M_{j}\in \mathrm {Subt}(c_{w,r}(\sigma ))_{u_{j}}. In addition, for every j\in \vert u \vert ,M_{j}\in \left(\left(\!\begin{}v^{\varphi (w_{i})}_{i}\\ f^{\sharp }_{w_{i}}[[W_{w_{i},l_{i}}]_{\Theta _{w_{i}}}]\end{}\!\... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
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935795feb1b97278240ac8ebb67c516a52d8c9bb | subsection | 133 | 174 | Tree Homomorphisms | Either (b.1.i) there exists a unique x\in X_{w_{i}} such that W_{i} = x or (b.1.ii) there exists a unique w^{\prime }\in S^{\star } a unique \nu \in \Sigma _{w^{\prime },w_{i}}, and a unique ((Q_{i^{\prime }})_{i^{\prime }\in \vert w^{\prime } \vert })\in \mathrm {T}_{\Sigma }(X)_{w^{\prime }} such that W_{i}=\nu ((Q_{... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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5428e3125253b9af1e2224f6f494ba7453632609 | subsection | 134 | 174 | Tree Homomorphisms | Hence, by the linearity of the tree homomorphism f^{\sharp }, we have that\left(\!\begin{}v^{\varphi (w^{\prime }_{i^{\prime }})}_{i^{\prime }}\\ f^{\sharp }_{w^{\prime }_{i^{\prime }}}(Q_{i^{\prime }})\end{}\!\right)_{i^{\prime }\in \vert w^{\prime } \vert }(\xi ((R_{j})_{j\in \vert u \vert }))=\xi \left(\left(\left(\... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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f0ed1ad44552a367acfee353393e4ad9b8040f5a | subsection | 135 | 174 | Tree Homomorphisms | Therefore, for every i^{\prime }\in \vert w^{\prime } \vert , there exists a \overline{Q}_{i^{\prime }}\in [Q_{i^{\prime }}]_{\Theta _{w^{\prime }_{i^{\prime }}}} such that N_{j}=\left(\!\begin{}v^{\varphi (w^{\prime }_{i^{\prime }})}_{i^{\prime }}\\ f^{\sharp }_{w^{\prime }_{i^{\prime }}}(\overline{Q}_{i^{\prime }})\e... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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78fbb5dac5b2ad101caa1d0e0432718b00f6a398 | subsection | 136 | 174 | Tree Homomorphisms | Again, by the linearity of the tree homomorphism f^{\sharp }, it follows, from Equation REF , that\xi ((M_{j})_{j\in \vert u \vert })=\xi \left(\left(\left(\!\begin{}v^{\varphi (w_{i})}_{i}\\ f^{\sharp }_{w_{i}}(Q_{i})\end{}\!\right)_{i\in \vert w \vert }(R_{j})\right)_{j\in \vert u \vert }\right).Hence, for every j\in... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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ea5bcfa674aa9880de88bf2ba7450c75e165bcfd | subsection | 137 | 174 | Tree Homomorphisms | By construction of \Phi , the set \lbrace f_{s}(x)\rbrace is \Phi _{\varphi (s)}-saturated. Moreover, since \Psi _{\varphi (s)} is a refinement of \Phi _{\varphi (s)}, we conclude that \lbrace f_{s}(x)\rbrace is \Psi _{\varphi (s)}-saturated. In this case, sinece M is related to f_{s}(x) for \Psi _{\varphi (s)}, we con... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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7e981e5a691d18d6f9ff684c5ea4e0774a48882c | subsection | 138 | 174 | Tree Homomorphisms | Since Q_{i}\in [P_{i}]_{\Theta _{w_{i}}}, and \Theta is a congruence on \mathrm {T}_{\Sigma }(X), we conclude that \sigma ((P_{i})_{i\in \vert w \vert }) and \sigma ((Q_{i})_{i\in \vert w \vert }) are \Theta _{s}-related. Moreover, since L is \Theta _{s}-saturated and \sigma ((P_{i})_{i\in \vert w \vert }) is a term in... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1201/9781482276367-14",
"end": 838,
"openalex_id": "https://openalex.org/W2911295106",
"raw": "F. Gécseg and M. Steinby, Tree languages. In G. Rozenberg and A. Salomaa, editors. Handbook of formal languages. Vol. 3. Beyond words, Chapter... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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e09f3d82309789516d3a04684da7d2aa680cbad4 | subsection | 139 | 174 | Tree Homomorphisms | Indeed, for ((\mathrm {id}_{S},c),f\circ \eta _{X}), where c = (c_{w,s})_{(w,s)\in S^{\star }\times S} is the family of mappings defined, for every (w,s)\in S^{\star }\times S, as follows:c_{w,s}
\left\lbrace
\begin{array}{@{\:}c@{\:}c@{\:}l}
\Sigma _{w,s} &\usebox {
}& \mathrm {T}_{\Sigma }(Y\cup \downarrow \! w)_{s... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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6f0227789c3cb447b2d98621828fb337a257868d | subsection | 140 | 174 | Tree Homomorphisms | If L\in \mathrm {Rec}_{s}(\mathbf {T}_{\Sigma }(X)), then f_{s}[L]\in \mathrm {Rec}_{s}(\mathbf {T}_{\Sigma }(Y)).We next provide, among other things, a categorial rendering of the just indicated result, but for a suitable class of homomorphisms.Proposition 3.30 Let \mathrm {Rec}_{\Sigma } be the mapping that sends an ... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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cff0b20b280197949ac7045462145527bc556fa3 | subsection | 141 | 174 | Tree Homomorphisms | Then, for every (w,s)\in S^{\star }\times S and every \sigma \in \Sigma _{w,s}, f^{@}_{s}[\cdot ]\circ \sigma ^{\wp } = \sigma ^{\wp }\circ f^{@}_{w}[\cdot ], i.e., for every (L_{i})_{i\in \vert w \vert }\in \prod _{i\in \vert w \vert }\mathrm {Rec}_{w_{i}}(\mathbf {T}_{\Sigma }(X)), the sets\textstyle (f^{@}_{s}[\cdot... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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2fa0c9e77c9e472198330379614a92ceaa09e092 | subsection | 142 | 174 | Tree Homomorphisms | Then the following diagram commutes@C=60pt{
\mathbf {T}_{\Sigma }(X)[r]^-{\lbrace \cdot \rbrace _{X}^{\sharp }} [d]_-{f^{@}}&
\mathbf {Rec}_{{\cdot }}(\mathbf {T}_{\Sigma }(X))[d]^-{(f^{@})^{\wp }}\\
\mathbf {T}_{\Sigma }(Y)[r]_-{\lbrace \cdot \rbrace _{Y}^{\sharp }}&
\mathbf {Rec}_{{\cdot }}(\mathbf {T}_{\Sigma }(Y))
... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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... |
083fbb7eac996d5977db6742c06f96ef8f03c833 | subsection | 143 | 174 | Tree Homomorphisms | Then the homomorphism (f^{@})^{\wp } =(f^{@}_{s}[\cdot ])_{s\in S} is such that, for every s\in S, f^{@}_{s}[\cdot ] is a Boolean algebra homomorphism from \mathbf {Rec}_{s}(\mathbf {T}_{\Sigma }(X)) to \mathbf {Rec}_{s}(\mathbf {T}_{\Sigma }(Y)).From the just stated proposition and Proposition REF we obtain the follow... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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-... |
865ed5305fc5f0c824e4a09efa156ff7c9690008 | subsection | 144 | 174 | Derivors and recognizability | Thatcher in , on p. 132, wrote: “Generally, the term `transformation' will mean any map from \mathrm {T}_{\Sigma } [the free algebra \mathrm {T}_{\Sigma }(\varnothing ), we add] into \mathrm {T}_{\Omega } [the free algebra \mathrm {T}_{\Omega }(\varnothing ), we add] where \Sigma = (\Sigma ,r) and \Omega = (\Omega ,s) ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/800169.805427",
"end": 341,
"openalex_id": "https://openalex.org/W2046933493",
"raw": "J. W. Thatcher, Transformations and translations from the point of view of generalized finite automata theory. In Proceedings of the ACM Symposiu... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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1adaf1b145a576dab11729c1a9bf5b457726abbe | subsection | 145 | 174 | Derivors and recognizability | Finally, after showing that every derivor is a hyperderivor, we state the counterparts of Propositions REF and REF for suitable morphisms of \mathbf {Alg}_{\mathfrak {d}}.Before defining the notion of Hall algebra, we recall that (1) a finitary specification is an ordered triple (S,\Sigma ,\mathcal {E}), where S is a s... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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... |
e931f7ae1cf0ebe60e7e1f2bc20f63e5c30be206 | subsection | 146 | 174 | Derivors and recognizability | A
Hall algebra for S is a \mathrm {H}_{S} =
(S^{\star }\times S,\Sigma ^{\mathrm {H}_{S}},\mathcal {E}^{\mathrm {H}_{S}})-algebra, where \Sigma ^{\mathrm {H}_{S}} is the S^{\star }\times S-sorted signature, i.e., the (S^{\star }\times S)^{\star }\times (S^{\star }\times S)-sorted set, defined as follows:For every w\in ... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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966aa42442e82f3609487bd8143af12e2fb1aaca | subsection | 147 | 174 | Derivors and recognizability | For every u, v, w\in S^{\star } and s\in S, the
equation
\xi _{u,v,s}(
\xi _{v,w,s}(v^{w,s}_{0},v^{v,w_{0}}_{1},\ldots ,
v^{v,w_{\vert w \vert -1}}_{\vert w \vert }),
v^{u,v_{0}}_{\vert w \vert +1},
\ldots ,v^{u,v_{\vert v \vert -1}}_{\vert w \vert +\vert v \vert }) = \\
\begin{aligned}\xi _{u,w,s}(v^{w,s}_{0},
&\xi _... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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f502108fe4ef71f664c6d9b7e9154f7a8e61f78a | subsection | 148 | 174 | Derivors and recognizability | Since \mathbf {Alg}(\mathrm {H}_{S}) is a variety, the forgetful functor \mathrm {G}_{\mathrm {H}_{S}} from \mathbf {Alg}(\mathrm {H}_{S}) to \mathbf {Set}^{S^{\star }\times S} has a left adjoint \mathbf {T}_{\mathrm {H}_{S}}, situation denoted by \mathbf {T}_{\mathrm {H}_{S}}\dashv \mathrm {G}_{\mathrm {H}_{S}}, or di... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/947864.947865",
"end": 1068,
"openalex_id": "https://openalex.org/W2062341323",
"raw": "J. Goguen and J. Meseguer, Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11(1985), pp. 307–334.",
"source_... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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698dcdf433c00549908894c93761da90214e9d0e | subsection | 149 | 174 | Derivors and recognizability | For every u,w\in S^{\star } and s\in S,
\xi _{u,w,s}^{\mathbf {Op}_{\mathrm {H}_{S}}(A)} is defined, for every f\in A _{s}^{ A_{w} } and g\in A^{A_{u}}_{w}, as
\xi _{u,w,s}^{\mathbf {Op}_{\mathrm {H}_{S}}(A)}(f,g_{0},\ldots ,g_{\vert w \vert -1})
= f\circ \langle g_{i}\rangle _{i\in \vert w \vert }, where \langle g_{i}... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.0... |
a3099994236b086576f749b7c7805725ee1989cf | subsection | 150 | 174 | Derivors and recognizability | Moreover, the finitary term operations on \mathbf {A} and the finitary algebraic operations on \mathbf {A} are subalgebras of the Hall algebra \mathbf {Op}_{\mathrm {H}_{S}}(\mathbf {A}).For every S-sorted signature \Sigma , \mathrm {Ter}_{\mathrm {H}_{S}}(\Sigma ) = (\mathrm {T}_{\Sigma }(\mathbin {\downarrow }w)_{s})... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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af50689a8c12724df28f583b87046efcbca05ae7 | subsection | 151 | 174 | Derivors and recognizability | For every u,w\in S^{\star } and s\in S,
\xi _{u,w,s}^{\mathbf {Ter}_{\mathrm {H}_{S}}(\Sigma )}
is the mapping
\xi _{u,w,s}^{\mathbf {Ter}_{\mathrm {H}_{S}}(\Sigma )}\left\lbrace
\begin{array}{@{\:}c@{\:}c@{\:}l}
\mathrm {T}_{\Sigma }(\mathbin {\downarrow }w)_{s} \times \mathrm {T}_{\Sigma }(\mathbin {\downarrow }u)_... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
-0.026034032925963402,
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... |
81b4bb47c4d7a9f244fcc5550d1a9671b3713e76 | subsection | 152 | 174 | Derivors and recognizability | Sometimes, to abbreviate, we will write \xi _{u,w,s} instead of
\xi _{u,w,s}^{\mathbf {Ter}_{\mathrm {H}_{S}}(\Sigma )}.Then \mathbf {Ter}_{\mathrm {H}_{S}}(\Sigma ) is a Hall algebra, the Hall algebra for (S,\Sigma ).Remark For every \Sigma -algebra \mathbf {A} there exists a homomorphism from the Hall algebra \mathbf... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.014596194960176945,
0.018434086814522743,
0.0350... |
68ba18d37926f9878eaccb160b69467edee838ad | subsection | 153 | 174 | Derivors and recognizability | Then, for every f\colon \Sigma \usebox {
}A and u\in S^{\star }, \mathbf {A}^{f,u}, the derived \Sigma -algebra of \mathbf {A} for (f,u),
is the \Sigma -algebra with underlying S-sorted set A^{f,u} = (A_{u,s})_{s\in S} and algebraic structure F^{f,u}, defined, for every (w,s)\in S^{\star }\times S, asF^{f,u}_{w,s} \lef... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.0... |
b1fa91fcda63842a8f1e7e96c912ed3e4033f3cd | subsection | 154 | 174 | Derivors and recognizability | Besides, for every u\in S^{\star }, we have that \mathbf {B}^{B_{u}}, the direct B_{u}-power of \mathbf {B}, is isomorphic to \mathbf {Op}_{\mathrm {H}_{S}}(B)^{G,u}.Lemma 3.37
Let \Sigma be an S-sorted signature, \mathbf {A} a Hall algebra, f\colon \Sigma \usebox {
}A and u\in S^{\star }. Then, for
every (w,s)\in S^{... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.... |
366d008fa7d2fc2d7261f28717484f951a588819 | subsection | 155 | 174 | Derivors and recognizability | Then we have that&(\sigma (Q_{0},\ldots ,Q_{\vert x \vert -1}))^{\mathbf {A}^{f,u}}
(a_{0},\ldots ,a_{\vert w \vert -1}) \\
&=
\sigma ^{\mathbf {A}^{f,u}}
(
Q_{0}^{\mathbf {A}^{f,u}}(a_{0},\ldots ,a_{\vert w \vert -1}),
\ldots ,
Q_{\vert x \vert -1}^{\mathbf {A}^{f,u}}(a_{0},\ldots ,a_{\vert w \vert -1})
)\\
&=
\xi _{u... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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-0.02282232977449894,
-0.000... |
a17316a61ef9554eff304bc9ab306c396e50e95e | subsection | 156 | 174 | Derivors and recognizability | Hypothesis)}
\end{aligned} \\
&=
\xi _{u,w,s}^{\mathbf {A}}
(
\xi _{w,x,s}^{\mathbf {A}}
(f(\sigma ),
(p^{w})^{\sharp }_{x_{0}}(Q_{0}),
\ldots ,
(p^{w})^{\sharp }_{x_{\vert x \vert -1}}(Q_{\vert x \vert -1})
),
a_{0},
\ldots ,
a_{\vert w \vert -1}
) \text{(by $\mathrm {H}_{3}$)} \\
&=
\xi _{u,w,s}^{\mathbf {A}}
(
\sigm... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.028213... |
62176a9384907febd8c1eebf8ddad0f0a5307aa7 | subsection | 157 | 174 | Derivors and recognizability | Let h be the S^{\star }\times S-sorted mapping defined, for every
(w,s)\in S^{\star }\times S, ash_{w,s}\left\lbrace
\begin{array}{@{\:}c@{\:}c@{\:}l}
\Sigma _{w,s} &\usebox {
}& \mathrm {T}_{\Sigma }(\mathbin {\downarrow }w)_{s} \\
\sigma &\longmapsto & \sigma \left(v^{w_{0}}_{0},\ldots ,v^{w_{\vert w \vert -1}}_{\v... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.0... |
ccc714f534c9113fcd5de0d7c82d12fc9ae81f2e | subsection | 158 | 174 | Derivors and recognizability | Then \widehat{f} is a homomorphism of Hall algebras, because, on the one hand, for w\in S^{\star } and i\in \vert w \vert we have that\widehat{f}_{w,w_{i}}((\pi ^{w}_{i})^{\mathbf {Ter}_{\mathrm {H}_{S}}(\Sigma )})
&=
\widehat{f}_{w,w_{i}}(v^{w_{i}}_{i}) \\
&=
p^{w}_{w_{i}}(v^{w_{i}}_{i}) \\
&=
(\pi ^{w}_{i})^{\mathbf ... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.0... |
cae508fcdf50c878419b2af6bc0ffb0d5b59b853 | subsection | 159 | 174 | Derivors and recognizability | Furthermore, \widehat{f}\circ h = f, because, for every w\in S^{\star }, s\in S, and \sigma \in \Sigma _{w,s}, we have that\widehat{f}_{w,s}(h_{w,s}(\sigma )) &=
(p^{w})^{\sharp }_{s}(\sigma (v^{w_{0}}_{0},\ldots ,v^{w_{\vert w \vert -1}}_{\vert w \vert -1})) \\
&=
\sigma ^{\mathbf {A}_{w}}
(p^{w}_{{w}_{0}}(v^{w_{0}}_{... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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-0.001843065372668206... |
d80c9eadcfce466369a0686f584c9447518597ae | subsection | 160 | 174 | Derivors and recognizability | Actually, (1) the mapping that assigns, for an S-sorted set A, to a structure of \Sigma -algebra F on A (i.e., an S^{\star }\times S-sorted mapping F from \Sigma to \mathrm {Op}_{\mathrm {H}_{S}}(A)) the homomorphism of Hall algebras \mathrm {Tr}^{(A,F)} = (\mathrm {Tr}^{\mathbin {\downarrow }w,(A,F)}_{s})_{(w,s)\in S^... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.030740372836589813,
... |
43bcbe472e22d963742b52bb295f8254abde2396 | subsection | 161 | 174 | Derivors and recognizability | The mappings that assign to operation symbols of a signature terms relative to another signature, together with mappings between the corresponding sets of sorts, form a new class of morphisms denominated derivors.Definition 3.39 Let \mathbf {\Sigma } = (S,\Sigma ) and \mathbf {\Lambda } = (T,\Lambda ) be many-sorted si... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1067,
"openalex_id": "https://openalex.org/W227694331",
"raw": "J. Climent Vidal and J. Soliveres Tur, A 2-categorical framework for the syntax and semantics of many-sorted equational logic. Rep. Math. Logic 45 (2010), 37–95.",
... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.005400863941758871,
... |
d987566c8da158bbe1720e56727e803e1dd95b86 | subsection | 162 | 174 | Derivors and recognizability | Since by,
Proposition REF , \mathbf {Ter}_{\mathrm {H}_{T}}(\Lambda ) is isomorphic to \mathbf {T}_{\mathrm {H}_{T}}(\Lambda ), the derivors can be defined, alternative, but equivalently, as ordered pairs \mathbf {d} = (\varphi ,d) with d\colon \Sigma \usebox {
}\mathrm {T}_{\mathrm {H}_{T}}(\Lambda ).On the other hand... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.... |
e5471bd0444a609f0b6ff64a780d5ebf29851126 | subsection | 163 | 174 | Derivors and recognizability | Therefore, each mapping \varphi \colon S\usebox {
}T between sets of sorts, determines a functor (\varphi ^{\star }\times \varphi , h^{\varphi })^{\ast } from \mathbf {Alg}(\mathrm {H}_{T}) to \mathbf {Alg}(\mathrm {H}_{S}), which transforms Hall algebras for T into Hall algebras for S. The action of the functor on the... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
0.010431555099785328,
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0.010546020232141018,
0... |
e028b68ca90168d0b05d6cd5468c59bcc29bb014 | subsection | 164 | 174 | Derivors and recognizability | Then \mathbf {e}\circ \mathbf {d} = (\psi ,e)\circ (\varphi ,d), the composition of \mathbf {d} and \mathbf {e}, is the derivor (\psi \circ \varphi ,e^{\sharp }_{\varphi ^{\star }\times \varphi }\circ d), where e^{\sharp }_{\varphi ^{\star }\times \varphi }\circ d is obtained from\begin{aligned}@C=40pt@R=30pt{
\Lambda ... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.035667672753334045,
-0.031579162925481796,
0.0... |
9de29d5922d68a0559b5d3e68e418d0ca4207c40 | subsection | 165 | 174 | Derivors and recognizability | In fact, for every set of sorts S, we have the adjoint situation \mathbf {T}_{\mathrm {H}_{S}}\dashv \mathrm {G}_{\mathrm {H}_{S}}
and, thus, a monad on \mathbf {Sig}(S) which we will denote by \mathbb {T}_{\mathrm {H}_{S}} = (\mathrm {T}_{\mathrm {H}_{S}},\eta ^{\mathrm {H}_{S}},\mu ^{\mathrm {H}_{S}}). Then the order... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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... |
169c4d263ce5535837c73a5917b6a9a6af51b300 | subsection | 166 | 174 | Derivors and recognizability | Moreover, by defining a suitable notion of transformation between derivors one can equip the category \mathbf {Sig}_{\mathfrak {d}} with a structure of 2-category (this was, in fact, already done in for polyderivors).Remark Since \mathbf {Sig} has coproducts, \mathbf {Kl}({\mathfrak {d}})\cong \mathbf {Sig}_{\mathfrak... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 218,
"openalex_id": "https://openalex.org/W227694331",
"raw": "J. Climent Vidal and J. Soliveres Tur, A 2-categorical framework for the syntax and semantics of many-sorted equational logic. Rep. Math. Logic 45 (2010), 37–95.",
... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
-0.04044025391340256,
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... |
d16a47066c87046f887c13c379bcab08c92de451 | subsection | 167 | 174 | Derivors and recognizability | Then \mathrm {Alg}_{\mathfrak {d}}(\mathbf {d}) is the functor from \mathbf {Alg}(\mathbf {\Lambda }) to \mathbf {Alg}(\mathbf {\Sigma }) that sends (B,G) to (B_{\varphi },G^{\mathbf {d}}) and a homomorphism f from (B,G) to (B^{\prime },G^{\prime }) to the homomorphism f_{\varphi } from (B_{\varphi },G^{\mathbf {d}}) t... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.008032017387449741,
0.025598717853426933,
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0.00418000528588891,
-0.002801290014758706... |
fff9a1c9a758af4d24227cb9c3dcdb3a60913c01 | subsection | 168 | 174 | Derivors and recognizability | Moreover, we have that \mathrm {Op}_{\mathrm {H}_{T}}(B)_{\varphi ^{\star }\times \varphi } = \mathrm {Op}_{\mathrm {H}_{S}}(B_{\varphi }) since, for every (w,s)\in S^{\star }\times S it holds that(\mathrm {Op}_{\mathrm {H}_{T}}(B)_{\varphi ^{\star }\times \varphi })_{w,s} &=
\mathrm {Op}_{\mathrm {H}_{T}}(B)_{\varphi ... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.03680187836289406,
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0.06676825135946274,
0... |
f31a4f4d1bb2f37c3ede44f0b89153ab951a0c3e | subsection | 169 | 174 | Derivors and recognizability | Moreover, (g\circ f)_{\varphi }=g_{\varphi }\circ f_{\varphi }, so that \mathrm {Alg}_{\mathfrak {d}}(\mathbf {d}) is a functor.From the definition of the functor \mathrm {Alg}_{\mathfrak {d}}, for every derivor \mathbf {d}\colon \mathbf {\Sigma }\usebox {
}\mathbf {\Lambda }, it is obvious that the following diagram@C... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
-0.029800090938806534,
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0.06396415084600449,
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0.008613493293523788,
-0.0002577276900410652,
0... |
c80c08504b54a99a78c40347264237db9f72f13b | subsection | 170 | 174 | Derivors and recognizability | Moreover, we have that{F^{(\psi ,e)}}^{(\varphi ,d)}
&=
(F^{\sharp }_{\psi ^{\star }\times \psi }\circ e)^{(\varphi ,d)} \\
&=
(F^{\sharp }_{\psi ^{\star }\times \psi }\circ e)^{\sharp }_{\varphi ^{\star }\times \varphi }\circ d \\
&=
({F^{\sharp }_{\psi ^{\star }\times \psi }}_{\varphi ^{\star }\times \varphi }\circ e... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
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d343bdd0c201d7c0b34b2c0e7d80db20039940fb | subsection | 171 | 174 | Derivors and recognizability | Finally, if f is a homomorphism of (U,\Omega )-algebras, then {f_{\psi }}_{\varphi } = f_{\psi \circ \varphi }.Definition 3.46 The category \mathbf {Alg}_{\mathfrak {d}} is \int ^{\mathbf {Sig}_{\mathfrak {d}}}\mathrm {Alg}_{\mathfrak {d}}, i.e., the category obtained by means of the Grothendieck construction applied t... | {
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... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
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9a226c6039a0e0db0947fd1402fb014735d0a6e5 | subsection | 172 | 174 | Derivors and recognizability | Then (\mathbf {d}^{Y},f) = ((\varphi ,d^{Y}),f), where d^{Y} is, for every (w,s)\in S^{\star }\times S, the composition of d_{w,s}, which is a mapping from \Sigma _{w,s} to \mathrm {T}_{\Lambda }(\downarrow \!\varphi ^{\star }(w))_{\varphi (s)}, and (\mathrm {in}^{@}_{\downarrow \varphi ^{\star }(w), Y\cup \downarrow \... | {
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} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
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] | [
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e54fccd75f3539dc99f3b69849071aa851ca9186 | subsection | 173 | 174 | Derivors and recognizability | Then \left(f^{\sharp }_{\varphi (s)}\right)^{-1}[L]\in \mathrm {Rec}_{s}(\mathbf {T}_{\Sigma }(X)).It follows from Proposition REF .Assumption For the following proposition, as was the case with Proposition REF , we will assume that S, T, \Sigma and X are finite.Proposition 3.49 Let \mathbf {d} be a linear derivor from... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
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bd9fc7c7cbc5c068594521a261281c5529f38ae5 | abstract | 0 | 63 | Abstract | The scalar-tensor theories have become popular recently in particular in
connection with attempts to explain present accelerated expansion of the
universe, but they have been considered as a natural extension of general
relativity long time ago. The Horndeski scalar-tensor theory involving four
invariantly defined Lagr... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
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d92b392b637c62546375656a4f96a2a98933a935 | subsection | 1 | 63 | Introduction | The modifications and generalizations of Einstein's theory of gravitation have been studied very actively in recent decades primarily in cosmological contexts with an attempt to explain the present accelerated expansions of the universe (for reviews, see, e.g., Refs. pap-sotiriou). The motivation for such studies is al... | {
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"doi": "10.1007/bf01807638",
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"raw": "G. W. Horndeski, “Second-order scalar-tensor field equations in a four-dimensional space,” Int. J. Theor. Phys. 10, 363-384 (1974)",
"source_r... | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
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bbc69c860afc4684cd4e13b90ed09dceea5ef075 | subsection | 2 | 63 | Introduction | Correspondingly, the covariant derivatives, the Christoffel symbols, and the curvature tensors are denoted by \nabla _{\alpha }, \Gamma ^{\lambda }_{\mu \nu }, R{^{\lambda }}_{\tau \rho \sigma } and by \bar{\nabla }_{\alpha }, \bar{\Gamma }^{\lambda }_{\mu \nu }, \bar{R}{^{\lambda }}_{\tau \rho \sigma }, the difference... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
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cb7d31c0b0cf215935a7aa099efe595c9953ce56 | subsection | 3 | 63 | Introduction | Assuming then that Einstein's equations and contracted Bianchi identities are satisfied one finds by straightforward though not short calculations that there exists a conserved vector density \hat{I}^{\alpha } equal to the divergence of a superpotential \hat{I}^{\alpha \beta }, \hat{I}^{\alpha } = \partial _{\beta } \h... | {
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{
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"raw": "A. N. Petrov and R. R. Lompay, “Covariantized Noether identities and conservation laws for perturbations in metric theories of gravity,” Gen. Relativ. Gravit. 45, 545-579 (2013)",
"source_ref... | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
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