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Lemma 1 For any connected graph \( G, v \in V\left( G\right) \) and \( {d}_{G}\left( v\right) = 1\left( {\;\operatorname{mod}\;2}\right) \), if \( G - v \) is up-embeddable, then \( G \) is also up-embeddable. In other words, if \( G \) is not up-embeddable, then \( G - v \) is also not up-embeddable. | Proof As \( G - v \) is up-embeddable, \( \xi \left( {G - v}\right) \leq 1 \) . Let \( T \) be the spanning tree of \( G - v \) such that \( \xi \left( {G - v}\right) = \xi \left( {G - v, T}\right) \leq 1 \) . By connecting \( v \) to \( T \) with an edge we obtain a spanning tree \( {T}^{\prime } \) of \( G \) such th... | Yes |
Lemma 2 The double dumb bell graphs are not up-embeddable. | Proof Let \( G \) be a double dumb bell graph, \( \left\{ {{w}_{1},{w}_{2},{w}_{3},{w}_{4}}\right\} \cup \left\{ {{v}_{1},{v}_{2},{v}_{3},{v}_{4}}\right\} \subseteq V\left( G\right) \) , \( \left\{ {{w}_{1},{w}_{2},{w}_{3},{w}_{4}}\right\} \cap \left\{ {{v}_{1},{v}_{2},{v}_{3},{v}_{4}}\right\} = \varnothing \), and \( ... | Yes |
Lemma 3 Let \( G \) be a 2-edge connected 3-regular graph as shown in Fig. 3(a), where \( {st} \) is the multiple edges of \( G \) . If \( G \) is not up-embeddable, then \( G \smallsetminus \{ s, t\} \) is also not up-embeddable. | Proof By using Lemma 1 twice, the result holds. | No |
Lemma 5 Let \( G \) be a 2-edge connected 3-regular graph as shown in Fig. 5, where \( \{ e, f\} \) is an edge-cut set of \( G \), and \( s, t \in V\left( {G}_{1}\right) ,\{ s, t\} \cap \{ u, v\} = \varnothing \) and \( {st} \) is the multiple edges of \( G \) . If \( G \smallsetminus \{ s, t\} \) is not 2-edge connect... | The proofs of Lemmas 4 and 5 can be found in [2]. | No |
Lemma 6 Let \( G \) be a 3-edge connected 3-regular graph \( \left( {G \neq {\theta }_{1}}\right) \) . Then, there exists one vertex \( x \in V\left( G\right) \) such that \( G - x \) is 2-edge connected. | Proof Assume that the edge \( e \) is a cut edge of \( G - x \), then there exists one component of \( G - x - e \) which has only one edge \( f \) connecting to the vertex \( x \) . Clearly, \( \{ e, f\} \) is an edge-cut of \( G \) . This is a contradiction. So, \( G - x \) must be 2-edge connected. | No |
Lemma 1.1 Let \( G \) be a bipartite graph and \( G = G\left\lbrack {V\left( {C}_{4}\right) \cup V\left( {P}_{2}\right) }\right\rbrack \) . If \( e\left( {{C}_{4},{P}_{2}}\right) \geq 3 \) , then \( G \) contains a \( {C}_{6} \) . | Proof Let \( {C}_{4} = \left\{ {{u}_{1},{v}_{1},{u}_{2},{v}_{2}}\right\} ,{P}_{2} = \left\{ {{x}_{1},{y}_{1}}\right\} \) . Assume that \( {x}_{1}{v}_{1},{y}_{1}{u}_{1} \in E \), then \( {x}_{1}{v}_{1}{u}_{2}{v}_{2}{u}_{1}{y}_{1}{x}_{1} \) is a 6-cycle. | Yes |
Lemma 2.1 Let \( w \) be any weight and \( \beta \geq 0 \) . Then for any measurable function \( f \) and each cube \( Q \subset {\mathbb{R}}^{n} \) , \[ {\left( \left( f - {m}_{f, w}\left( Q\right) \right) {\chi }_{Q}\right) }_{w}^{ * }\left( t\right) \leq \frac{2}{{\lambda }_{n}\log 2}w{\left( Q\right) }^{\beta }{\be... | Proof Just in the same way as the proof of Theorem 0.1, applying Theorem 1.1 to \( f - \) \( {m}_{f, w}\left( Q\right) \), we get \[ {\left( \left( f - {m}_{f, w}\left( Q\right) \right) {\chi }_{Q}\right) }_{w}^{ * }\left( t\right) \leq {2w}{\left( Q\right) }^{\beta }{\begin{Vmatrix}{M}_{{\lambda }_{n}, w}^{\# ,\beta }... | Yes |
Lemma 2.2 Let \( w \) be any weight and \( \beta \geq 0 \) . Then for \( 0 < \lambda \leq {\lambda }_{n} \) , \[ \lambda \parallel {M}_{\lambda, w}^{\# ,\beta }f{\parallel }_{\infty } \leq \parallel f{\parallel }_{{L}_{\beta }\left( w\right) } \leq c\parallel {M}_{\lambda, w}^{\# ,\beta }f{\parallel }_{\infty }. \] | Proof We only need to prove the second inequality. For any cube \( Q \), by Lemma 2.1, we have \[ \frac{1}{w{\left( Q\right) }^{1 + \beta }}{\int }_{Q}\left| {f\left( x\right) - {f}_{Q, w}}\right| w\left( y\right) \mathrm{d}y \leq \frac{2}{w{\left( Q\right) }^{1 + \beta }}{\int }_{0}^{w\left( Q\right) }{\left( \left( f... | Yes |
Lemma 2.3 Let \( f \geq 0 \) and let \( \left\{ {Q}_{\varepsilon }\right\} \) be a family of cubes, containing a cube \( Q \), such that \( {Q}_{\varepsilon } \subset {Q}_{\delta } \) when \( \varepsilon < \delta \) and \( {Q}_{\varepsilon } \rightarrow Q \) as \( \varepsilon \rightarrow 0 \) . Then\n\n\[ \lim \mathop{... | Proof By the properties of the median value,\n\n\[ \left| {{\left( f{\chi }_{Q}\right) }_{w}^{ * }\left( \frac{w\left( Q\right) }{2}\right) - {\left( f{\chi }_{{Q}_{\varepsilon }}\right) }_{w}^{ * }\left( \frac{w\left( {Q}_{\varepsilon }\right) }{2}\right) }\right| \]\n\n\[ \leq {\left( {\left( f - f{\chi }_{Q}\right) ... | Yes |
Lemma 2.4 Let \( f \geq 0 \) . For \( \alpha ,\delta > 0 \), let \( \Omega = \left\{ {x \in Q : w{\left( Q\right) }^{\beta }{M}_{\frac{1}{2}, w}^{\# ,\beta }f\left( x\right) > \alpha }\right\} \) and \( E = \{ x \in Q : {\widetilde{m}}_{w}f\left( x\right) > \delta \} \) . Suppose that \( w{\left( Q\right) }^{-\beta }{\... | Proof For any \( x \in E \smallsetminus \Omega \), let\n\n\[ \begin{array}{l} {r}_{x} = \sup \left\{ {r \in (0,{l}_{Q}\rbrack : w{\left( \widetilde{Q}\right) }^{-\beta }{\left( f{\chi }_{\widetilde{Q}\left( {x, r}\right) }\right) }_{w}^{ * }\left( \frac{\widetilde{Q}\left( {x, r}\right) }{2}\right) > \delta }\right\} .... | Yes |
Theorem 0.2 Let \( R = \inf \{ {R}_{0} > 0\;|\;\Omega \subset {B}_{G}\left( {R}_{0}\right) \} . \) If \( 1 \leq k < \frac{{2n} + 1}{2} \) and \( f \in {W}^{1,\infty }\left( \Omega \right) \cap \) \( {C}_{0}^{1}\left( \Omega \right) \), then for all \( \xi = r{\xi }^{ * } \in \Omega \) and \( F \in {W}^{1,\infty }\left(... | \[ \left| {f\left( \xi \right) - \frac{1}{\left| \Omega \right| }{\int }_{\Omega }f\left( \eta \right) \mathrm{d}\eta }\right| \leq \mathcal{N}\left( F\right) + \left( {1 - \frac{\left| \Omega \right| }{\left| {B}_{G}\left( R\right) \right| }}\right) \left| {\frac{1}{\left| \Omega \right| }{\int }_{\Omega }f\left( \eta... | Yes |
Lemma 1.1 Assume that \( \gamma > - {2n} \), and\n\n\[ \n{C}_{\gamma } = {\int }_{\sum }{\left| {z}^{ * }\right| }^{\gamma }\mathrm{d}\mu \n\]\n\nThen\n\n\[ \n{C}_{\gamma } = \frac{2{\pi }^{n}\Gamma \left( \frac{{2n} + \gamma }{4k}\right) \Gamma \left( \frac{1}{2}\right) }{\Gamma \left( n\right) \Gamma \left( \frac{Q +... | Proof For any \( \gamma > - Q \), from (1.1) it holds\n\n\[ \n{C}_{\gamma } = {\int }_{\sum }{\left| {z}^{ * }\right| }^{\gamma }\mathrm{d}\mu \n\]\n\n\[ \n= \left( {Q + \gamma }\right) {\int }_{0}^{1}{r}^{Q + \gamma - 1}\mathrm{\;d}r{\int }_{\sum }{\left| {z}^{ * }\right| }^{\gamma }\mathrm{d}\mu \n\]\n\n\[ \n= \left(... | Yes |
Lemma 2.1 \( \begin{aligned} & \text{ Assume that }0 < {R}_{1} < {R}_{2} < \infty ,1 \leq k < \frac{{2n} + 1}{2} \\ & \text{ and }f \in {C}^{1}\left( \overline{{B}_{G}\left( {R}_{2}\right) \smallsetminus {B}_{G}\left( {R}_{1}\right) }\right) . \end{aligned} \) Then\n\n\[ \left| {{\int }_{\sum }f\left( {{R}_{2}{\xi }^{ ... | Proof Applying the pointwise Schwartz inequality to (1.5), it follows that\n\n\[ \left| {{\int }_{\sum }f\left( {{R}_{2}{\xi }^{ * }}\right) \mathrm{d}\mu - {\int }_{\sum }f\left( {{R}_{1}{\xi }^{ * }}\right) \mathrm{d}\mu }\right| \leq {\int }_{{B}_{G}\left( {R}_{2}\right) \smallsetminus {B}_{G}\left( {R}_{1}\right) }... | Yes |
For any integer \( k \) with \( 1 \leq k \leq n - 1 \) we denote by \( {H}^{n - k} = \{ \left( {v, w}\right) \in \) \( {\mathbb{R}}_{1}^{n - k + 1}\;|\;\upsilon \cdot \upsilon - {\omega }^{2} = - 1,\omega > 0\} \; \) the hyperbolic space embedded in the Minkowski space \( \;{\mathbb{R}}_{1}^{n - k + 1}. \) We define \(... | In [4], the authors have proved that the Laguerre second fundamental form of \( x \) is parallel, and \( x \) is a Laguerre isoparametric hypersurface. | Yes |
For any positive integers \( {m}_{1},{m}_{2},\cdots ,{m}_{s} \) with \( {m}_{1} + {m}_{2} + \cdots + {m}_{s} = n - 1 \) and any non-zero constants \( {\lambda }_{1},{\lambda }_{2},\cdots ,{\lambda }_{s} \), we define \( x : {\mathbb{R}}^{n - 1} \rightarrow {\mathbb{R}}_{0}^{n} \), a space-like oriented hypersurface in ... | In [4], Li et al. have proved that the Laguerre second fundamental form of \( x \) is parallel, and \( x \) is a Laguerre isoparametric hypersurface. | No |
Theorem 1.2 Two umbilical free oriented hypersurfaces in \( {\mathbb{R}}^{n}\left( {n > 3}\right) \) with non-zero principal curvatures are Laguerre equivalent if and only if they have the same Laguerre metric \( g \) and Laguerre second fundamental form \( B \) . | We define \( {\widetilde{E}}_{i} = {r}_{i}{e}_{i},1 \leq i \leq n - 1 \), then \( \left\{ {{\widetilde{E}}_{1},{\widetilde{E}}_{2},\cdots ,{\widetilde{E}}_{n - 1}}\right\} \) is an orthonormal basis for III \( = \mathrm{d}\xi \cdot \mathrm{d}\xi \) . Then \( \left\{ {{E}_{i} = {\rho }^{-1}{\widetilde{E}}_{i} \mid 1 \le... | Yes |
Lemma 2.1 \( {B}_{{13},4}{B}_{{23},4} = 0 \) . | Proof From (2.4),(2.5) and the totally symmetric of \( {B}_{{ij}, k} \), we have\n\n\[ \n{\Gamma }_{1k}^{2} = {\Gamma }_{2k}^{1} = 0,\;k = 3,4;\;{\Gamma }_{ik}^{i} = {\Gamma }_{ii}^{k} = 0,\;\text{ if }i \neq k\text{ and }\left( {i, k}\right) \neq \left( {1,2}\right) ,\left( {2,1}\right) .\n\]\n\n(2.6)\n\nThen, by (2.3... | Yes |
Lemma 2.2 \( {B}_{{13},4} \) is constant. | Proof From (2.5) and (1.7), we have\n\n\[ \mathop{\sum }\limits_{k}{B}_{{13},{4k}}{w}_{k} = \mathrm{d}{B}_{{13},4} + \mathop{\sum }\limits_{k}\left( {{B}_{{k3},4}{w}_{k1} + {B}_{{1k},4}{w}_{k3} + {B}_{{13}, k}{w}_{k4}}\right) = \mathrm{d}{B}_{{13},4}, \]\n\n(2.15)\n\nFrom (2.8) and (2.12), we get \( {B}_{{13},{42}} = 0... | Yes |
Lemma 2.3 If \( {B}_{{13},4} \neq 0 \), then the followings hold:\n\n\[ \n{R}_{1313} = \frac{2{B}_{{13},4}^{2}}{\left( {{b}_{4} - {b}_{3}}\right) \left( {{b}_{4} - {b}_{1}}\right) },\;{R}_{1414} = \frac{2{B}_{{13},4}^{2}}{\left( {{b}_{3} - {b}_{1}}\right) \left( {{b}_{3} - {b}_{4}}\right) }\n\]\n\n\[ \n{R}_{3434} = \fr... | Proof From (2.17) and (2.18), we get\n\n\[ \n{B}_{{13},{31}} = {B}_{{33},{11}} = 2{B}_{{13},4}{\Gamma }_{13}^{4},\;{B}_{{13},{13}} = {B}_{{11},{33}} = 2{B}_{{13},4}{\Gamma }_{31}^{4}.\n\]\n\n(2.19)\n\nUsing (2.4),(2.10) and (2.19), we obtain \( {R}_{1313} = \frac{2{B}_{13}^{2}{}_{4}}{\left( {{b}_{4} - {b}_{3}}\right) \... | Yes |
Lemma 2.4 Denote by \( i, j, k, l \) the four distinct elements of \( \{ 1,2,3,4\} \) with order arbitrary given, then we have\n\n\[ \n{R}_{{ij},{ij}} = \frac{2{B}_{{ij}, k}^{2}}{\left( {{b}_{i} - {b}_{k}}\right) \left( {{b}_{j} - {b}_{k}}\right) } + \frac{2{B}_{{ij}, l}^{2}}{\left( {{b}_{i} - {b}_{l}}\right) \left( {{... | Proof Using the similar proof as that of Lemma 2.3. | No |
Lemma 1.1 Assume that (H4) holds. Then RBDSDEs (0.2) have a solution. Moreover, there are a minimal and a maximal solutions to RBDSDEs (0.2). | Proof We define \( {f}_{n}\left( {t, y, z}\right) = \mathop{\inf }\limits_{{\left( {u, v}\right) \in {Q}^{1 + d}}}\{ f\left( {t, y, z}\right) + n\left| {y - u}\right| + n\left| {z - v}\right| \} \) for \( n \geq C \) . For every \( n \geq C,{f}_{n} \) is uniformly \( n \) -Lipschitz and \( \left( {f}_{n}\right) \) conv... | No |
Lemma 2.1 Under the assumptions (H5)-(H7), for all \( n = 1,2,\cdots \), and \( t \in \left\lbrack {0, T}\right\rbrack \) ,(2.1) has a minimal solution \( \left( {{\underline{y}}_{t}^{n},{\underline{z}}_{t}^{n},{\underline{k}}_{t}^{n}}\right) \in {S}^{2}\left( {0, T;\mathbb{R}}\right) \times {M}^{2}\left( {0, T;{\mathb... | Proof For \( n = 1 \), we consider the following RBDSDEs:\n\n\[ \begin{cases} {\underline{y}}_{t}^{1} = & \xi + {\int }_{t}^{T}\left( {f\left( {s,{Y}_{s}^{1},{Z}_{s}^{1}}\right) + h\left( {s,{\underline{y}}_{s}^{1} - {Y}_{s}^{1},{\underline{z}}_{s}^{1} - {Z}_{s}^{1}}\right) }\right) \mathrm{d}s \\ & + {\underline{k}}_{... | Yes |
Theorem 1.2 Let \( \mathcal{T} \) be a positive random variable, independent of \( \left\{ {{X}_{i} : i \geq 1}\right\} \), with regularly varying tail distribution with parameter \( \lambda \in \lbrack 0,1) \), i.e., \( P\left( {\mathcal{T} > t}\right) = L\left( t\right) {t}^{-\lambda } \), where \( L\left( \cdot \rig... | (2) If \( \theta \in \left( {0,\infty }\right) \), then \[ P\left( {\mathop{\max }\limits_{\substack{{0 < k \leq \mathcal{T}N} \\ {0 < L \leq N} }}{\xi }_{L}^{\left( N\right) }\left( k\right) > u}\right) \sim \Gamma \left( {1 - \lambda }\right) P\left( {\mathcal{T} > {\left( {J}_{\theta }{u}^{2}\Psi \left( u\right) \ri... | Yes |
Lemma 2.2 Let \( {\left( {\xi }_{k}\right) }_{k = 1}^{\infty } \) be a sequence of uniformly bounded random variables, i.e., there exists some \( M \in \left( {0,\infty }\right) \) such that \( \left| {\xi }_{k}\right| \leq M \) a.s. for all \( k \in \mathbb{N} \) . If\n\n\[ \operatorname{Var}\left( {\mathop{\sum }\lim... | Proof See [7, Lemma 3.1]. | No |
Theorem 4. \( {1}^{\left\lbrack {24}\right\rbrack }\; \) For \( \;\gamma \in {\mathcal{P}}_{\tau }\left( {2n}\right) \), let \( P = \gamma \left( \tau \right) \) . If \( {i}_{{L}_{0}}\left( \gamma \right) \geq 0,\;{i}_{{L}_{1}}\left( \gamma \right) \geq 0,\;i\left( \gamma \right) \geq n \) , \( {\gamma }^{2}\left( t\ri... | \[ {i}_{{L}_{1}}\left( \gamma \right) + {S}_{{P}^{2}}^{ + }\left( 1\right) - {\nu }_{{L}_{0}}\left( \gamma \right) \geq 0 \] where \( {S}_{M}^{ + }\left( 1\right) \) is the splitting number of the symplectic matrix \( M \) (see [25]). | No |
Lemma 2.1 Let \( \mathcal{D} \) be a \( 2 - \left( {v,{19},1}\right) \) design admitting a block-transitive but not flag-transitive automorphism group \( G \) . Then \( v = {342}{b}_{2} + 1 \) . | Proof Since \( k = {19} \) and \( {k}_{1} = \left( {k, v}\right) ,{k}_{1} = 1 \) or 19. If \( {k}_{1} = {19} \), then \( k \mid v \), by [12], \( G \) is flag-transitive, a contradiction. Hence, we have \( {k}_{1} = 1 \) . Thus \( v = k\left( {k - 1}\right) {b}_{2} + 1 = {342}{b}_{2} + 1 \) . | Yes |
Lemma 2.2 Let \( \mathcal{D} \) be a \( 2 - \left( {v,{19},1}\right) \) design admitting a block-transitive but not flag-transitive automorphism group \( G \) and \( \operatorname{Soc}\left( G\right) = T \) be even order. If \( G \) is non-solvable, then \( \left| T\right| \leq {172}{\left| {T}_{\alpha }\right| }^{2}\l... | Proof Let \( B = \{ 1,2,\cdots ,{19}\} \in \mathcal{B} \), and \( \lambda ,\theta \) be the length of the longest suborbit of \( G \) and \( T \) on \( \mathcal{P} \), respectively. Since \( G \) is block-transitive but not flag-transitive, we have \( \theta \leq \left| {T}_{\alpha }\right| \), where \( \alpha \in \mat... | Yes |
Example 1.1 Consider the ring \( \bar{R} = \left\{ {\left( \begin{array}{ll} a & b \\ 0 & c \end{array}\right) \mid a, b, c \in R}\right\} \) with \( R \) a reduced ring. Let \( \beta : \bar{R} \rightarrow \bar{R} \) be an endomorphism of \( \bar{R} \) defined by\n\n\[ \beta \left( \left( \begin{array}{ll} a & b \\ 0 &... | Then we obtain\n\n\[ f\left( x\right) \beta \left( {g\left( x\right) }\right) = \left( \begin{matrix} {a}_{0}{d}_{0} & 0 \\ 0 & 0 \end{matrix}\right) + \left( \begin{matrix} {a}_{0}{d}_{1} + {a}_{1}{d}_{0} & 0 \\ 0 & 0 \end{matrix}\right) x + \left( \begin{matrix} {a}_{1}{d}_{1} & 0 \\ 0 & 0 \end{matrix}\right) {x}^{2}... | Yes |
Example 1.2 Let \( F \) be a field and consider a ring \( R = \left( \begin{array}{ll} F & F \\ 0 & F \end{array}\right) \). Let\n\n\[ f\left( x\right) = \left( \begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right) + \left( \begin{matrix} - 1 & - 1 \\ 0 & 0 \end{matrix}\right) x,\;g\left( x\right) = \left( \begin{array}{... | \[ \left( \begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right) \alpha \left( \left( \begin{array}{ll} 0 & - 1 \\ 0 & - 1 \end{array}\right) \right) = \left( \begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right) \left( \begin{matrix} 0 & 1 \\ 0 & - 1 \end{matrix}\right) \neq 0, \] | Yes |
Proposition 1.1 Let \( I \) be a reduced ideal (as a ring without identity) of a ring \( R \) . If \( R/I \) is a right \( \alpha \) -weak Armendariz ring, then \( R \) is right \( \alpha \) -weak Armendariz. | Proof Note that if \( {ab} = 0 \) with \( a, b \in R \), then we have \( {bIa} = 0 \) since \( {\left( bIa\right) }^{2} = 0 \) and \( I \) is reduced. We freely use this fact in the following. Let \( f\left( x\right) = {a}_{0} + {a}_{1}x, g\left( x\right) = {b}_{0} + {b}_{1}x \in R\left\lbrack x\right\rbrack \) with \(... | Yes |
Proposition 1.4 Let \( R \) be a ring with an endomorphism \( \alpha \) . Then \( R \) is right \( \alpha \) -weak Armendariz if and only if \( {\bigtriangleup }^{-1}R \) is right \( \alpha \) -weak Armendariz. | Proof It suffices to show the necessity. Assume that \( R \) is right weak \( \alpha \) -Armendariz. Let \( f\left( x\right) = {u}_{0}^{-1}{a}_{0} + {u}_{1}^{-1}{a}_{1}x, g\left( x\right) = {v}_{0}^{-1}{b}_{0} + {v}_{1}^{-1}{b}_{1}x \in {\bigtriangleup }^{-1}R\left\lbrack x\right\rbrack \) with \( f\left( x\right) \alp... | Yes |
Proposition 1.5 Let \( R \) be a reduced ring. If \( {M}_{R} \) is a semicommutative module, then \( {M}_{R} \) is an \( \alpha \) -weak McCoy module. | Proof Suppose that \( \alpha \left( {m\left( x\right) }\right) g\left( x\right) = 0 \) with \( m\left( x\right) = {m}_{0} + {m}_{1}x \in M\left\lbrack x\right\rbrack \) and \( g\left( x\right) = {a}_{0} + \) \( {a}_{1}x \in R\left\lbrack x\right\rbrack \smallsetminus \{ 0\} \) . We may assume \( {a}_{0} \neq 0 \), then... | Yes |
Proposition 2.4 If \( R \) is a reduced ring, then the ring \( R\left\lbrack x\right\rbrack /\left( {x}^{3}\right) \) is right \( \alpha \) -weak quasi-Armendariz, where \( \left( {x}^{3}\right) \) is the two-sided ideal of \( R\left\lbrack x\right\rbrack \) generated by \( {x}^{3} \) . | Proof Since \( R\left\lbrack x\right\rbrack /\left( {x}^{n}\right) \cong {V}_{n - 1} \), the result follows directly from Proposition 2.3. | No |
Proposition 2.5 Let \( R \) be a ring \( R \) with an endomorphism \( \alpha \) . Then \( R\left\lbrack x\right\rbrack \) is right \( \alpha \) -weak quasi-Armendariz if and only if \( R\left\lbrack {x;{x}^{-1}}\right\rbrack \) is right \( \alpha \) -weak quasi-Armendariz. | Proof It is straightforward. | No |
Proposition 2.6 Let \( R \) be a right Ore ring with \( \alpha \) an epimorphism of \( R \), and \( Q\left( R\right) \) the classical right quotient ring of \( R \) . Then \( R \) is right \( \alpha \) -weak quasi-Armendariz if and only if \( Q\left( R\right) \) is right \( \bar{\alpha } \) -weak quasi-Armendariz. | Proof It suffices to show the necessity. Let \( f\left( x\right) = {\alpha }_{0} + {\alpha }_{1}x, g\left( x\right) = {\beta }_{0} + {\beta }_{1}x \in Q\left\lbrack x\right\rbrack \) with \( f\left( x\right) q\left( x\right) \alpha \left( {g\left( x\right) }\right) = 0 \) for every \( q\left( x\right) = {\gamma }_{0} +... | Yes |
Proposition 1.3 \( {}^{\left\lbrack 8,\text{ Theorem 1.29 }\right\rbrack } \) Let \( R \) be an alternating matrix on field \( {\mathbb{F}}_{2} \), namely, \( R \) is skew symmetric (or symmetric because of \( {\mathbb{F}}_{2} \) ) and the entries on its principal diagonal are all 0 . Then there exists an invertible ma... | \[ {AR}{A}^{\mathrm{T}} = \operatorname{diag}\left( {\left( \begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) ,\left( \begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) ,\cdots ,\left( \begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) ,0,0,\cdots ,0}\right) . \] | No |
Proposition 1.4 \( {}^{\left\lbrack 8,\text{ Theorem 1.6 }\right\rbrack } \) Let \( {\Phi }_{n} = {\operatorname{gl}}_{n}\left( {\mathbb{F}}_{2}\right) \) denote the general linear group of order \( n\left( { \geq 1}\right) \) on the finite field \( {\mathbb{F}}_{2} \) . Then the number of elements in \( {\Phi }_{n} \)... | \[ \left| {\Phi }_{n}\right| = {2}^{\frac{n\left( {n - 1}\right) }{2}}\mathop{\prod }\limits_{{i = 1}}^{n}\left( {{2}^{i} - 1}\right) . \] | Yes |
Theorem 2.2 The group operation defined in Definition 2.2 and the group operation defined in Definition 2.3 are equivalent. | Proof For any \( P \in \mathcal{Q} \), we can suppose \( P = {c}_{f} + \operatorname{RM}\left( {1, m - 1}\right) \), where \( f \) is a homogeneous Boolean function of degree 2 such that \( f = \mathop{\sum }\limits_{{1 \leq i < j \leq m - 1}}{b}_{ij}{x}_{i}{x}_{j} \) . Let \( {b}_{ii} = 0,{b}_{ji} = {b}_{ij} \) for \(... | Yes |
Theorem 2.4 Notations are defined as Theorems 2.1 and 2.3. Then, for \( 0 \leq k \leq \lfloor \frac{m - 1}{2}\rfloor \) , the weight enumerator of coset \( {P}_{k} \) is\n\n\[ \n{W}_{{P}_{k}}\left( y\right) = {2}^{2k}{y}^{{2}^{m - 2} - {2}^{m - k - 2}} + \left( {{2}^{m} - {2}^{{2k} + 1}}\right) {y}^{{2}^{m - 2}} + {2}^... | Proof (i) First, for coset \( {P}_{0} = \operatorname{RM}\left( {1, m - 1}\right) \), which is a 1st-order RM code, it is known that\n\n\[ \n{W}_{{P}_{0}}\left( y\right) = 1 + \left( {{2}^{m} - 2}\right) {y}^{{2}^{m - 2}} + {y}^{{2}^{m - 1}}. \n\]\n\n(ii) Second, we prove that the theorem holds in the special case \( m... | Yes |
Theorem 2.5 Notations are defined as Theorems 2.1 and 2.3. For \( 0 \leq k \leq \left\lfloor \frac{m - 1}{2}\right\rfloor \), we have\n\n\[ \left| {H}_{{B}_{k, m - 1}}\right| = {2}^{{k}^{2} + {2k}\left( {m - {2k} - 1}\right) }\mathop{\prod }\limits_{{i = 1}}^{k}\left( {{4}^{i} - 1}\right) \cdot \left| {\Phi }_{m - {2k}... | In order to prove Theorem 2.5, we first show the following two lemmas.\n\nLemma 2.1 For \( t \geq | No |
Lemma 1.3 Let \( I \) be a regular ideal of a ring \( R \) . Then the following are equivalent:\n\n(1) \( I \) is Hermite.\n\n(2) For any \( x \in {I}^{2} \), there exists \( y \in {\operatorname{col}}_{2}\left( R\right) \) such that \( x = {xyx} \) .\n\n(3) For any \( x \in {}^{2}I \), there exists \( y \in {\operator... | Proof (1) \( \Rightarrow \) (3) For any \( x \in {}^{2}I \), it follows from Lemma 1.2 that there exists a \( U \in \) \( {\mathrm{{GL}}}_{2}\left( R\right) \) such that \( \left( {x,0}\right) = \left( {x,0}\right) U\left( {x,0}\right) \) . Let \( y \in {R}^{2} \) be the first row of \( U \) . Then \( x = {xyx} \), as ... | Yes |
Theorem 1.1 Let \( I \) be a regular ideal of a ring \( R \) . Then the following are equivalent:\n\n(1) \( I \) is Hermite.\n\n(2) For any \( x \in {I}^{2} \), there exists \( y \in {\operatorname{col}}_{2}\left( R\right) \) such that \( {xy} \in I \) is an idempotent.\n\n(3) For any \( x \in {}^{2}I \), there exists ... | Proof \( \;\left( 1\right) \Rightarrow \left( 3\right) \; \) For each \( x \in {I}^{2} \), it follows from Lemma 1.3 that there exists \( y \in {}^{2}R \) which is a column of some invertible matrix such that \( x = {xyx} \) . Hence \( {xy} \in I \) is an idempotent, as required.\n\n(3) \( \Rightarrow \) (1) Given any ... | Yes |
Corollary 1.1 Let \( I \) be a regular ideal of a ring \( R \) . If \( {eRe} \) is Hermite for all idempotents \( e \in I \), then \( I \) is Hermite. | Proof For any \( x \in {I}^{2} \), we write \( x = \left( {{x}_{1},{x}_{2}}\right) \), where \( {x}_{1},{x}_{2} \in I \) . As \( I \) is regular, there exists an idempotent \( e \in I \) such that \( {x}_{1},{x}_{2} \in {eRe} \) . Hence, \( x \in {\left( eRe\right) }^{2} \) . By hypothesis, there exists \( y \in {\oper... | Yes |
Theorem 1.2 Let \( I \) be a regular ideal of a ring \( R \) . Then the following are equivalent:\n\n(1) \( I \) is Hermite.\n\n(2) \( a\left( {{}^{2}R}\right) + {bR} = R \) with \( a \in {I}^{2}, b \in R \) implies that there exists \( y \in {R}^{2} \) such that \( a + {by} \in \) \( {\operatorname{row}}_{2}\left( R\r... | Proof \( \left( 1\right) \Rightarrow \left( 2\right) \;a\left( {{}^{2}R}\right) + {bR} = R \) with \( a \in {I}^{2}, b \in R \) implies that there exist \( x \in {}^{2}R \) , \( y \in R \) such that \( {ax} + {by} = 1 \) . In view of Theorem 1.1, there exists a \( u \in {\operatorname{row}}_{2}\left( R\right) \) such t... | No |
Corollary 1.2 Let \( I \) be a regular ideal of a ring \( R \) . Then the following are equivalent:\n\n(1) \( I \) is Hermite.\n\n(2) For any idempotents \( e \in I \) and \( f \in {M}_{2}\left( I\right) ,\varphi : {eR} \cong f\left( {2R}\right) \) implies that there exists \( u \in {\operatorname{col}}_{2}\left( R\rig... | Proof \( \left( 1\right) \Rightarrow \left( 2\right) \; \) For any idempotents \( e \in I \) and \( f \in {M}_{2}\left( I\right) ,\varphi : {eR} \cong f\left( {2R}\right) \) implies that there exist some \( {r}_{1},{r}_{2} \in R \) such that \( f{e}_{1} = \varphi \left( {e{r}_{1}}\right) \) and \( f{e}_{2} = \varphi \l... | Yes |
Example 1.1 Let \( R \) be a regular, right self-injective ring, and let \( e \in R, f = \left( {f}_{ij}\right) \in {M}_{2}\left( R\right) \) be idempotents. If \( {eR},{f}_{ij}R \) are all directly finite, then \( \varphi : {eR} \cong f\left( {2R}\right) \) implies that there exists \( u \in {\operatorname{col}}_{2}\l... | Proof Let \( H = \{ x \in R \mid {xR} \) is a directly finite right \( R \) -module \( \} \) . Then \( H \) is an ideal of \( R \) . For any idempotent \( e \in H \) , \( {eR} \) is a directly finite right \( R \) -module, and so \( {\operatorname{End}}_{R}\left( {eR}\right) \cong {eRe} \) is a directly finite ring. He... | No |
Lemma 2.1 Let \( I \) be a regular ideal of a ring \( R \) . Then the following are equivalent:\n\n(1) \( I \) is Hermite.\n\n(2) For any \( m, n \in \mathbb{N}\left( {m \geq n + 1}\right) ,{mR} = {A}_{1} \oplus {B}_{1},{nR} = {A}_{2} \oplus {B}_{2} \) with \( {A}_{1} \cong {A}_{2},{A}_{1} = \) \( {A}_{1}I,{A}_{2} = {A... | Proof (1) \( \Rightarrow \) (2) Suppose that \( {mR} = {A}_{1} \oplus {B}_{1},{nR} = {A}_{2} \oplus {B}_{2} \) with \( {A}_{1} \cong {A}_{2},{A}_{1} = \) \( {A}_{1}I,{A}_{2} = {A}_{2}I \) . Then there exists a regular homomorphism \( f : {mR} \rightarrow {nR} \) such that \( \ker f \cong {B}_{1} \) , \( \operatorname{I... | Yes |
Proposition 2.1 Let \( I \) be a regular ideal of a ring \( R \) . Then the following are equivalent:\n\n(1) \( I \) is Hermite.\n\n(2) For any \( m, n \in \mathbb{N}\left( {m \geq n + 1}\right) \) and any \( X \in {M}_{m \times n}\left( I\right) \), there exists a completable \( U \in {M}_{n \times m}\left( R\right) \... | Proof (1) \( \Rightarrow \) (2) Since \( I \) is regular, we see that \( X \in {M}_{m \times n}\left( I\right) \) is regular. Hence, there exists \( Y \in {M}_{n \times m}\left( R\right) \) such that \( X = {XYX} \) . Clearly, \( {mR} = {XY}\left( {mR}\right) \oplus \left( {{I}_{m} - {XY}}\right) \left( {mR}\right) \) ... | Yes |
Corollary 2.1 Let \( I \) be a regular ideal of a ring \( R \) . Then the following are equivalent:\n\n(1) \( I \) is Hermite.\n\n(2) For any \( m, n \in \mathbb{N}\left( {m \geq n + 1}\right) \) and any \( X \in {M}_{m \times n}\left( I\right) \) ,\n\n\[ \left( {X,{\mathbf{0}}_{m \times \left( {m - n}\right) }}\right)... | Proof This is obvious from Proposition 2.1. | No |
Lemma 2.2 Every regular \( B \) -ideal of a ring is Hermite. | Proof Let \( I \) be a regular \( B \) -ideal of \( R \) . For any idempotent \( e \in I \), one easily checks that \( {eRe} \) is unit-regular. Hence, \( {eRe} \) is Hermite. Therefore, \( I \) is Hermite by Corollary 1.1. | No |
Theorem 2.1 Let \( R \) be a ring, and let \( A = \left( {a}_{ij}\right) \in {M}_{m \times n}\left( R\right) \) . If each \( R{a}_{ij}R \) is a regular \( B \) -ideal, then \( A \) can be completed to a unit-regular matrix. | Proof Set \( I = \{ x \in R \mid {RxR} \) is a regular \( B \) -ideal \( \} \) . Let \( x, y \in I \) and \( z \in R \) . Then \( {RxzR} \subseteq {RxR} \) . Hence, \( {RxzR} \) is a \( B \) -ideal. Let \( t \in {RxzR} \) . Then we have some \( s \in {RxR} \) such that \( t = {tst} = t\left( {sts}\right) t \) . Since \... | Yes |
Corollary 2.2 Let \( R \) be a unit-regular ring, and let \( A = \left( {a}_{ij}\right) \in {M}_{m \times n}\left( R\right) \). Then \( A \) can be completed to a unit-regular matrix. | Proof Since \( R{a}_{ij}R \) is an ideal of \( R \), we see that each \( R{a}_{ij}R \) is a regular \( B \) -ideal of \( R \). This completes the proof. | No |
Example 2.1 Let \( R \) be a regular ring, and let \( A = \left( {a}_{ij}\right) \in {M}_{m \times n}\left( R\right) \). If for any \( r \in R \), each \( 1 + {a}_{ij}r \in R \) is unit-regular, then \( A \) can be completed to a unit-regular matrix. | Proof Let \( \Psi \left( R\right) = \{ x \in R \mid 1 + {xr} \in R \) is unit-regular for all \( r \in R\} \). In terms of \( \lbrack 4 \) , Lemma 13.2.24], \( \Psi \left( R\right) \) is a \( B \) -ideal of \( R \). Therefore, we complete the proof by Lemma 2.2. | No |
Example 2.2 Let \( R \) be a regular ring, and let \( A = \left( {a}_{ij}\right) \in {M}_{m \times n}\left( R\right) \) . If \( R{a}_{ij}R \) is of bounded index, then \( A \) can be completed to a unit-regular matrix. | Proof Let \( I = \{ x \in R \mid {RxR} \) is of bounded index \( \} \) . By virtue of \( \left\lbrack {4\text{, Example 13.2.20}}\right\rbrack, I \) is a strongly \( \pi \) -regular ideal of \( R \) . For any idempotent \( e \in I \) , \( {eRe} \) is strongly \( \pi \) -regular; hence, it has stable range one. This imp... | No |
Lemma 2.3 Let \( I \) be a regular ideal of a ring \( R \) . Then the following are equivalent:\n\n(1) \( I \) is Hermite.\n\n(2) For any \( m, n \in \mathbb{N}\left( {m \geq n + 1}\right) \) and any \( X \in {M}_{m \times n}\left( I\right) \), there exists a completable \( U \in {M}_{n \times m}\left( R\right) \) such... | Proof \( \;\left( 1\right) \Rightarrow \left( 2\right) \; \) For any \( m, n \in \mathbb{N}\;\left( {m \geq n + 1}\right) \) and any \( X \in {M}_{m \times n}\left( I\right) \), there exists a completable \( U \in {M}_{n \times m}\left( R\right) \) such that \( X = {XUX} \) . Let \( {XU} = E \) . Then \( E = {E}^{2} \i... | Yes |
Proposition 2.2 Let \( I \) be a regular ideal of a ring \( R \) . Then the following are equivalent:\n\n(1) \( I \) is Hermite.\n\n(2) For any \( m, n \in \mathbb{N}\left( {m \geq n + 1}\right) \) and any \( X \in {M}_{m \times n}\left( I\right) \), there exist an idempotent \( E \in {M}_{m}\left( I\right) \) and a co... | Proof \( \left( 1\right) \Rightarrow \left( 2\right) \) For any \( m, n \in \mathbb{N}\left( {m \geq n + 1}\right) \) and any \( X \in {M}_{m \times n}\left( I\right) \), it follows from Proposition 2.1 that there exists a completable \( V \in {M}_{n \times m}\left( R\right) \) such that \( X = {XVX} \) . Assume that \... | Yes |
Corollary 2.2 Let \( R \) be a ring, and let \( A = \left( {a}_{ij}\right) \in {M}_{m \times n}\left( R\right) \) . If each \( R{a}_{ij}R \) is a regular \( B \) -ideal of \( R \), then there exist an idempotent \( E \in {M}_{m}\left( R\right) \) and a completable \( U \in {M}_{m \times n}\left( R\right) \) such that \... | Proof Set \( I = \{ x \in R \mid {RxR} \) is a regular \( B \) -ideal of \( R\} \) . As in the proof of Theorem 2.1, we see that \( I \) is a regular \( B \) -ideal of \( R \) . Clearly, \( A \in {M}_{m \times n}\left( I\right) \) . Therefore, we complete the proof in terms of Proposition 2.2. | No |
Lemma 2.4 Every strongly separative regular ideal of a ring is Hermite. | Proof Let \( I \) be a strongly separative regular ideal of a ring \( R \), and let \( e \in I \) . Then \( {eRe} \) is strongly separative. Clearly, \( {eRe} \) is regular. Hence, \( {eRe} \) is Hermite by [4, Theorem 14.3.1]. Therefore, \( I \) is Hermite in terms of Corollary 1.1. | Yes |
Theorem 2.2 Let \( R \) be a regular ring, and let \( A = \left( {a}_{ij}\right) \in {M}_{m \times n}\left( R\right) \) . If each \( {\operatorname{End}}_{R}\left( {{a}_{ij}R}\right) \) is strongly separative, then there exist an idempotent \( E \in {M}_{m}\left( R\right) \) and a completable \( U \in \) \( {M}_{m \tim... | Proof Let \( I = \left\{ {a \in R \mid {\operatorname{End}}_{R}\left( {aR}\right) }\right. \) is strongly separative \( \} \) . In light of \( \left\lbrack {4\text{, Lemma 14.4.6}}\right\rbrack \) , \( I \) is a strongly separative ideal of \( R \) . It follows from Lemma 2.4 that \( I \) is an Hermite ideal of \( R \)... | Yes |
Corollary 2.3 Let \( R \) be a regular ring, and let \( A = \left( {a}_{ij}\right) \in {M}_{m \times n}\left( R\right) \) . If each \( {\operatorname{End}}_{R}\left( {{a}_{ij}R}\right) \) is unit-regular, then there exist an idempotent \( E \in {M}_{m}\left( R\right) \) and a completable \( U \in {M}_{m \times n}\left(... | Proof This is obvious by Theorem 2.2. | No |
Lemma 2 \( G \) has no adjacent 2-vertices. | Proof Assume the contrary that \( {uv} \in E\left( G\right), d\left( u\right) = d\left( v\right) = 2 \) . By the minimality of \( G \) , \( G - {uv} \) has a \( \left( {\Delta + 5}\right) - 2 \) -distance coloring \( \varphi \) . We discolor \( u \) and \( v \), then we get \( \left| {F\left( u\right) }\right| \leq \De... | Yes |
Lemma 3 (1) If \( v = \left( {{v}_{1},{v}_{2},{v}_{3}}\right) \), where \( d\left( {v}_{1}\right) = 2 \), then \( d\left( {v}_{2}\right) + d\left( {v}_{3}\right) \geq \Delta + 4 \) . | Proof (1) Suppose to the contrary that \( d\left( {v}_{2}\right) + d\left( {v}_{3}\right) \leq \Delta + 3 \) and let \( {G}^{ * } = G - v{v}_{1} \) . By the minimality of \( G,{G}^{ * } \) has a \( \left( {\Delta + 5}\right) - 2 \) -distance coloring. Now we discolor \( v \) and \( {v}_{1} \) . Because of the inequalit... | Yes |
Lemma 4 Let 4-vertex \( v = \left( {{v}_{1},{v}_{2},{v}_{3},{v}_{4}}\right) \), (1) If \( d\left( {v}_{1}\right) = d\left( {v}_{2}\right) = 2 \), then \( d\left( {v}_{3}\right) + d\left( {v}_{4}\right) \geq \Delta + 3 \) . | Proof (1) Assume that \( d\left( {v}_{3}\right) + d\left( {v}_{4}\right) \leq \Delta + 2 \) and let \( {G}^{ * } = G - v{v}_{1} \) . By the minimality of \( G,{G}^{ * } \) has a \( \left( {\Delta + 5}\right) \) -2-distance coloring. Now we discolor \( v,{v}_{1} \) and \( {v}_{2} \) . Because of the inequality \( \left|... | Yes |
Lemma 5 Let 5-vertex \( v = \left( {{v}_{1},{v}_{2},{v}_{3},{v}_{4},{v}_{5}}\right) \). (1) If \( d\left( {v}_{1}\right) = d\left( {v}_{2}\right) = d\left( {v}_{3}\right) = d\left( {v}_{4}\right) = 2 \), then \( d\left( {v}_{5}\right) \geq \Delta - 3 \). (2) If \( d\left( {v}_{1}\right) = d\left( {v}_{2}\right) = d\lef... | Proof (1) Suppose that \( d\left( {v}_{5}\right) \leq \Delta - 4 \). Let \( {G}^{ * } = G - v{v}_{1} \). By the minimality of \( G \), \( {G}^{ * } \) has a \( \left( {\Delta + 5}\right) \) -2-distance coloring. Now we discolor \( {v}_{i}\left( {i = 1,2,3,4}\right) \) and \( v \). Because of the inequality \( \left| {F... | Yes |
Lemma 6 Let 6-vertex \( v = \left( {{v}_{1},{v}_{2},{v}_{3},{v}_{4},{v}_{5},{v}_{6}}\right) \) . If \( v \) is adjacent to six 2-vertices, then \( v \) is weak-adjacent to six \( \Delta \) -vertices. | Proof Suppose that \( {v}_{i}^{\prime } \) is another neighbor different from \( v \) of \( {v}_{i}\left( {i = 1,2,\cdots ,6}\right) \) . Assume that \( {v}_{j}\left( {j \in \{ 1,2,\cdots ,6\} }\right) \) satisfies \( d\left( {v}_{j}^{\prime }\right) \leq \Delta - 1 \) . Let \( {G}^{ * } = G - v{v}_{j} \) . By the mini... | Yes |
Lemma 7 Let 7-vertex \( v = \left( {{v}_{1},{v}_{2},{v}_{3},{v}_{4},{v}_{5},{v}_{6},{v}_{7}}\right) \) . If \( v \) is adjacent to seven 2-vertices, then \( v \) is weak-adjacent to seven \( {\left( \Delta - 1\right) }^{ + } \) -vertices. | Proof Suppose that \( {v}_{i}^{\prime } \) is another neighbor different from \( v \) of \( {v}_{i}\left( {i = 1,2,\cdots ,7}\right) \) . Assume that there is \( 1 \leq j \leq 7 \) satisfying \( d\left( {v}_{j}^{\prime }\right) \leq \Delta - 2 \) . Let \( {G}^{ * } = G - v{v}_{j} \) . By the minimality of \( G,{G}^{ * ... | Yes |
Lemma 8 Let 8-vertex \( v = \left( {{v}_{1},{v}_{2},{v}_{3},{v}_{4},{v}_{5},{v}_{6},{v}_{7},{v}_{8}}\right) \) . If \( v \) is adjacent to eight 2-vertices, then \( v \) is weak-adjacent to eight \( {\left( \Delta - 2\right) }^{ + } \) -vertices. | Proof Suppose that \( {v}_{i}^{\prime } \) is another neighbor different from \( v \) of \( {v}_{i}\left( {i = 1,2,\cdots ,8}\right) \) . Assume that there is a \( 1 \leq j \leq 8 \) satisfying \( d\left( {v}_{j}^{\prime }\right) \leq \Delta - 3 \) . Let \( {G}^{ * } = G - v{v}_{j} \) . By the minimality of \( G,{G}^{ ... | Yes |
Lemma 5 Let \( G = \left( {V, E}\right) \) be a 2-connected graph and \( {uv} \in E \) . If \( G = {G}_{1} \cup {G}_{2} \) and \( {G}_{1} \cap {G}_{2} \simeq G\left\lbrack {uv}\right\rbrack \), then \( {\gamma }_{\mathrm{{sc}}}^{\prime }\left( G\right) = \left| E\right| - 4 \) if and only if \( {G}_{1},{G}_{2} \in \lef... | Proof The sufficiency is obvious. Next we show the necessity. Let \( {\gamma }_{\mathrm{{sc}}}^{\prime }\left( G\right) = \left| E\right| - 4 \) and \( f \) be a minimum SCDF of \( G \) with \( f\left( {e}_{1}\right) = f\left( {e}_{2}\right) = - 1 \) . Then either one of \( \left\{ {{e}_{1},{e}_{2}}\right\} \) is \( {u... | Yes |
Theorem 1.1 Let \( C\left( x\right) \in {L}^{2}\left( {0,1}\right) \), then there exists a classical solution \( \left( {n, V}\right) \) to (1.3)- (1.6) such that \( n\left( x\right) \geq {\mathrm{e}}^{-M} > 0 \) for \( x \in \left( {0,1}\right) \), where \( M \) satisfies\n\n\[ M = \left| {\log {n}_{0}}\right| + \sqrt... | ## 2 Proof of Results\n\nWe first reformulate the system (1.3)-(1.4) as an elliptic fourth-order equation. In fact, when we divide (1.3) by \( n \) and differentiate with respect to \( x \) ,(1.3) is formally equivalent to\n\n\[ - {\varepsilon }^{2}{\left( \frac{{\left( \sqrt{n}\right) }_{xx}}{\sqrt{n}}\right) }_{xx} +... | Yes |
Lemma 1 When \( p > 1,{\lambda }_{1},{\lambda }_{2},{\lambda }_{3},{\lambda }_{4} > 0,{\lambda }_{3} + 2{\lambda }_{4} \geq {\lambda }_{2},\min \left\{ {{\lambda }_{3,}{\lambda }_{4}}\right\} \geq {\lambda }_{1}, t > q \geq 1 \) , then the following inequality holds:\n\n\[ 2{\lambda }_{1}{\left( t - q\right) }^{p} + {\... | Proof Let\n\n\[ f\left( t\right) = {\lambda }_{3}{t}^{p} + {\lambda }_{4}{\left( t + q\right) }^{p} - 2{\lambda }_{1}{\left( t - q\right) }^{p} - {\lambda }_{2}{q}^{p}, \]\n\nwhere \( p > 1,{\lambda }_{1},{\lambda }_{2},{\lambda }_{3},{\lambda }_{4} > 0,{\lambda }_{3} + 2{\lambda }_{4} \geq {\lambda }_{2},\min \left( {... | Yes |
Lemma 2 Let \( n > 1 \), and \( n \in \mathbb{N},{\lambda }_{1} \geq {\lambda }_{2} \geq {\lambda }_{3} > 0, t > q \geq 1 \) . Then the following inequality holds:\n\n\[ \n{\lambda }_{1}{\left( t + q\right) }^{n + 1} + {\lambda }_{2}{\left( t - q\right) }^{n + 1} > 2{\lambda }_{3}\left( {{t}^{n + 1} + {q}^{n + 1}}\righ... | Proof Let\n\n\[ \nf\left( t\right) = {\lambda }_{1}{\left( t + q\right) }^{n + 1} + {\lambda }_{2}{\left( t - q\right) }^{n + 1} - 2{\lambda }_{3}\left( {{t}^{n + 1} + {q}^{n + 1}}\right) ,\n\]\n\nwhere \( n > 1, n \in \mathbb{N},{\lambda }_{1} \geq {\lambda }_{2} \geq {\lambda }_{3} > 0, t > q \geq 1 \) .\n\nThen\n\n\... | Yes |
Theorem 2 Let \( X \) be a closed convex subset of \( E, D \) be a bounded open subset in \( X \) , and \( \theta \in D \) . Suppose that \( A : \Omega \times \bar{D} \rightarrow X \) is a random semi-closed 1-set-contractive operator, such that\n\n\[ \n{\lambda }_{1}{\begin{Vmatrix}A\left( \omega, x\right) + qx + p\en... | Proof Imitating the proof of Theorem 1, we can easily prove the above conclusion with Lemma 2. | No |
Theorem 3 Let \( X \) be a closed convex subset of \( E, D \) be a bounded open subset in \( X \) , and \( \theta \in D \) . Suppose that \( A : \Omega \times \bar{D} \rightarrow X \) is a random semi-closed 1-set-contractive operator, such that\n\n\[ \left( {\mathrm{W}}_{3}\right) \;\parallel A\left( {\omega, x}\right... | Proof We can obtain the theorem analogously too, so we omit its proof here. | No |
Theorem 1.1 \( {}^{\left\lbrack {13},\text{ Theorem 4.17 }\right\rbrack } \) If \( \beta \leq \delta \) and \( k + a \geq l + b \), then there are continuous embeddings:\n\n\[ \n{C}_{\beta }^{k + 1}\left( E\right) \subseteq {C}_{\beta }^{k, a}\left( E\right) \subseteq {C}_{\delta }^{l, b}\left( E\right) \subseteq {C}_{... | Proof Our method is derived from the proof of [11, Theorem 4.8]. In view of the second conclusion in [13, Theorem 4.2], we have the sequence of continue maps:\n\n\[ \n{C}_{\beta }^{k + 1}\left( E\right) \xrightarrow[]{{\mathrm{e}}^{\left( {r - s}\right) t}}{B}_{\beta }^{k + 1}\left( E\right) \rightarrow {B}_{\beta }^{k... | Yes |
Proposition 2.1 \( {}^{\left\lbrack {13},\text{ Proposition 6.39 }\right\rbrack } \) Let \( k \geq 2 \) and \( \alpha + 1 \in {\mathbb{R}}^{L} \) with \( \alpha + 1 < 1 \) . Then the map \( {F}_{\alpha + 1} : {V}_{\alpha + 1}^{k + 1, a} \rightarrow {C}_{\alpha }^{k, a}\left( X\right) \oplus {C}_{\alpha }^{k, a}\left( {... | \[ {F}_{\alpha + 1}^{\prime }\left( 0\right) : {C}_{\alpha + 1}^{k + 1, a}\left( {{T}^{ * }X}\right) \rightarrow {C}_{\alpha }^{k, a}\left( X\right) \oplus {C}_{\alpha }^{k, a}\left( {{\Lambda }^{2}{T}^{ * }X}\right) \] at 0 which acts as \( {\mathrm{d}}^{ * } + \mathrm{d} \) . | Yes |
Lemma 2.1 \( {}^{\left\lbrack 5\right\rbrack } \) Let \( \left( {M, F}\right) \) be an \( \left( {\alpha ,\beta }\right) \) -space. Then its fundamental tensor is\n\n\[ \n{g}_{ij} = \rho {a}_{ij} + {\rho }_{0}{b}_{i}{b}_{j} + {\rho }_{1}\left( {{b}_{i}{\alpha }_{j} + {b}_{j}{\alpha }_{i}}\right) + {\rho }_{2}{\alpha }_... | In order to calculate the inverse matrix of \( {g}_{ij} \) we can rewrite\n\n\[ \n{g}_{ij} = \rho \left( {{A}_{ij} + \mu {Y}_{i}{Y}_{j}}\right) \n\] \n\nwhere\n\n\[ \n{A}_{ij} \mathrel{\text{:=}} {a}_{ij} + \delta {b}_{i}{b}_{j},\;{Y}_{i} \mathrel{\text{:=}} {\alpha }_{i} + \varepsilon {b}_{i}, \n\] \n\n\[ \n\varepsilo... | Yes |
Lemma 2.3 Let \( \left( {{M}^{n}, F}\right) \) is an \( n \) -dimensional Finsler manifold and \( F = \frac{{\alpha }^{2}}{\beta } \) is a Kropina metric. Then the norm of mean Cartan torsion I at point \( x \in M \) is\n\n\[ \parallel \mathbf{I}{\parallel }_{x} = \frac{n + 1}{\sqrt{6\sqrt{3}}}\sqrt{b} \] | Proof For the general \( \left( {\alpha ,\beta }\right) \) -metric, by Lemma 2.2 we have\n\n\[ {I}_{i} = - \frac{\Phi }{2F\Delta }\left( {\phi - s{\phi }^{\prime }}\right) {h}_{i} \]\n\nwhere \( {h}_{i} = {b}_{i} - {\alpha }^{-1}s{y}_{i} \) . From (2) we get\n\n\[ {g}^{ij}{I}_{i}{I}_{j} = \frac{{\Phi }^{2}(\phi - s{\ph... | Yes |
Theorem 4.1 \( {}^{\left\lbrack {21}\right\rbrack } \) Let \( \left( {M, F}\right) \) be a forward complete Finsler space with bounded Cartan torsion. Suppose that \( F \) is R-quadratic, then \( F \) must be a Landsberg metric. In particular, every compact R-quadratic Finsler space must be Landsbergian. | Proof of Corollary 0.1 First by Theorems 0.1-0.2 we know that the Cartan torsion of \( F = \frac{{\alpha }^{m + 1}}{{\beta }^{m}}\left( {m \geq 1}\right) \) must be bounded when \( b \mathrel{\text{:=}} \parallel \beta {\parallel }_{\alpha } \) is bounded on \( M \) . According to Theorem 4.1, Shen \( {}^{\left\lbrack ... | No |
Lemma 0.3 Let \( {e}_{1} = {\left( 0,1,0,\cdots \right) }^{\mathrm{T}} \) and \( S{\left( {a}_{n}\right) }_{n = 0}^{\infty } = {\left( {a}_{n + 1}\right) }_{n = 0}^{\infty } \) . Then we have\ni) In \( B,\phi \in B \Leftrightarrow z \rightarrow \frac{{\phi }^{\prime }\left( z\right) - {\phi }^{\prime }\left( 0\right) }... | Proof \( \;i) \) For \( \phi \left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}^{n} \), we have \( {\phi }^{\prime }\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }\left( {n + 1}\right) {a}_{n + 1}{z}^{n} \) and \( \frac{{\phi }^{\prime }\left( z\right) - {\phi }^{\prime }\left( 0\right) }... | Yes |
1) \( \parallel {A\phi }\parallel \geq \parallel \phi \parallel ,\forall \phi \in D\left( A\right) \) . | Proof \( \;1) \) We see that \( \parallel {A\phi }{\parallel }^{2} = \mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {n + 1}\right) {\left| {\phi }_{n}\right| }^{2},\;\forall \phi = \{ {\phi }_{n}{\} }_{n = 1}^{\infty } \in D\left( A\right) \), then \( \parallel {A\phi }\parallel \geq \) \( \parallel \phi \parallel ,\f... | No |
Theorem 1.2 For \( \mu \neq 0 \), the Jacobi-Gribov matrix is proper, i.e., \( {H}_{\min } = {H}_{\max } \), and its spectrum \( \sigma \left( {H}_{\max }\right) \) is discrete. | Proof i) \( D\left( {H}_{\min }\right) \subset D\left( {H}_{\max }\right) \) is obvious.\n\nii) \( D\left( {H}_{\max }\right) \subset D\left( {H}_{\min }\right) \) . In fact, letting \( \psi = {\left( {\psi }_{n}\right) }_{n = 1}^{\infty } \in D\left( {H}_{\max }\right) \), according that \( {H}_{\min } \) is invertibl... | Yes |
Corollary 1.1 Let \( {\mathcal{P}}_{0} = \{ p \in \mathcal{P};p\left( 0\right) = 0\} \) . Then \( H\left\lbrack \mathcal{P}\right\rbrack \) is dense in \( {B}_{0} \) . | Proof Let \( \psi \in {B}_{0} \) . As \( 0 \in \rho \left( {H}_{\min }\right) \) and \( {H}_{\min } = {H}_{\max } \mathrel{\text{:=}} H \), then there exists \( \phi \in \) \( D\left( H\right) \mathrel{\text{:=}} D\left( {H}_{\max }\right) = D\left( {H}_{\min }\right) \) such that \( \psi = {H\phi } \) . Hence, there e... | Yes |
Theorem 1.3 \( {}^{\left\lbrack {12}\right\rbrack } \) For \( \mu = 0 \), the Jacobi-Gribov matrix is not proper, i.e., \( {H}_{\min } \neq {H}_{\max } \) and \( \sigma \left( {H}_{\max }\right) = \mathbb{C} \) . | In the case \( \mu = 0 \), the boundary conditions at infinity are used in a description of all maximal dissipative extensions of \( {H}_{\min } \) and the characteristic functions of these dissipative extensions are computed. Completeness theorems are obtained for the system of generalized eigenvectors. For details of... | No |
Theorem 2.1 \( {H}_{m}^{-1} \) is discrete approximation of \( {H}^{-1} \), i.e., i) There exists \( M > 0 \) such that \( {\begin{Vmatrix}{H}_{m}^{-1}\end{Vmatrix}}_{m} \leq M \) for \( m \) large enough; ii) \( \begin{Vmatrix}{{H}_{m}^{-1}\mathop{\prod }\limits_{m}\phi - \mathop{\prod }\limits_{m}{H}^{-1}\phi }\end{V... | Proof i) Since \( {H}^{-1} \) is a bounded operator, then there exists \( M > 0 \) such that \( {\begin{Vmatrix}{H}_{m}^{-1}\end{Vmatrix}}_{m} \leq \) \( M \) for \( m \) large enough. ii) Let \( \phi \left( z\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{\phi }_{n}{e}_{n}\left( z\right) \) . Then \[ \begin{Vmatri... | Yes |
Theorem 2.2 Let \( {\sigma }_{m} \) be a sequence of eigenvalues of \( {H}_{m} \) such that \( {\sigma }_{m} \rightarrow \sigma \) when \( m \rightarrow \infty . \) Then \( \sigma \) is an eigenvalue of \( H \) . | Proof Let \( {\phi }_{m} = {\left\{ {\phi }_{n}^{m}\right\} }_{n = 1}^{m} \) be an eigenvector of \( {H}_{m} \) associated to eigenvalue \( {\sigma }_{m} \) such that \( \begin{Vmatrix}{\phi }_{m}\end{Vmatrix} = 1 \) . Then we have \( {H}_{m}{\phi }_{m} = {\sigma }_{m}{\phi }_{m} \) and \( \Re \left\langle {{H}_{m}{\ph... | Yes |
Theorem 2.3 Let \( \mu > 0 \) and \( {P}_{m}\left( z\right) = \det \left( {{H}_{m} - {zI}}\right) \) . Then\ni) Two successive polynomials \( {P}_{m - 1}\left( z\right) \) and \( {P}_{m}\left( z\right) \) can not vanish simultaneously. | Proof i) If two successive polynomials vanish simultaneously, then according to (2.5) they would cancel all, which contradicts \( {P}_{0}\left( z\right) = 1 \) . | Yes |
Theorem 2.4 For \( \lambda < \frac{\mu }{2\sqrt{2}} \), the Jacobi-Gribov operator has a least real eigenvalue. | Proof By according vi) of Theorem 2.3, we can find a subsequence \( {\sigma }_{j} \) of the smallest zeros \( {x}_{{2k} + 1} \) of \( {P}_{{2k} + 1}\left( x\right) \) which converges to \( {\sigma }_{1} \in \left( {\mu ,{x}_{2}}\right) \) . And by Theorem 2.2, we deduce that \( {\sigma }_{1} \) is an eigenvalue of Jaco... | No |
Proposition 2.1 Let\n\n\\[ \n\\left\\{ \\begin{array}{l} {\\widetilde{P}}_{m + 1}\\left( z\\right) = \\frac{1}{{\\alpha }_{1}{\\alpha }_{2}\\cdots {\\alpha }_{m}}{P}_{m}\\left( z\\right) ,\\;m \\geq 1, \\\\ {\\widetilde{P}}_{m + 1}\\left( z\\right) = 0, \\\\ {\\widetilde{P}}_{1}\\left( z\\right) = 1. \\end{array}\\righ... | Proof Let \\( {z}_{0} \\) be an eigenvalue of \\( {H}_{m} \\) and let \\( {\\left( {\\phi }_{1},{\\phi }_{2},\\cdots ,{\\phi }_{m}\\right) }^{\\mathrm{T}} \\) be the corresponding\neigenvector. Then we have the system of linear equations\n\n\\[ \n\\left\\{ \\begin{array}{l} {q}_{1}{\\phi }_{1} + {\\alpha }_{1}{\\phi }_... | Yes |
Lemma 3.2 \( {P}_{1}\left( k\right) \) and \( {P}_{2}\left( k\right) \) defined by (3.6) and (3.7) admit the following asymptotical expansions as \( k \rightarrow \infty \) . | 1) \( {P}_{1}\left( k\right) = {a}_{0} + \frac{{a}_{1}}{k} + o\left( \frac{1}{k}\right) ,\;\mathrm{{with}}\;{a}_{0} = 2\;\mathrm{{and}}\;{a}_{1} = \frac{{\mu }^{2} - 4{\lambda }^{2}}{2{\lambda }^{2}}. \) | Yes |
Lemma 1.3 Let \( S = \left( {Y,{S}_{\alpha }}\right) \) be an ortho-lc-monoid. Then for any \( \alpha \in Y,{\mathcal{L}}^{*, \sim }\left( S\right) \cap \left( {{S}_{\alpha } \times }\right. \) \( \left. {S}_{\alpha }\right) = {\mathcal{L}}^{*, \sim }\left( {S}_{\alpha }\right) ,{\mathcal{R}}^{*, \sim }\left( S\right) ... | Proof It is clear that \( {\mathcal{L}}^{*, \sim }\left( S\right) \cap \left( {{S}_{\alpha } \times {S}_{\alpha }}\right) \subseteq {\mathcal{L}}^{*, \sim }\left( {S}_{\alpha }\right) \).\n\nSuppose that \( \left( {a, b}\right) \in {\mathcal{L}}^{*, \sim }\left( {S}_{\alpha }\right) \), and let \( e \in {H}_{a}^{*, \si... | Yes |
Lemma 1.5 Let \( S = I \times M \times \Lambda \) be a rectangular lc-monoid. If \( \rho \) is a congruence on \( S \) [a \( \left( {*, \sim }\right) \) -good congruence on \( S\rbrack \), then \( {\rho }_{M} \) is a congruence on \( M \) [a \( \left( {*, \sim }\right) \) -good congruence on \( M \) ] and \( {\rho }_{E... | Proof It is clear that \( {\rho }_{M} \) is a congruence on \( M \) and \( {\rho }_{E} \) is a congruence on \( E \) . We only need to prove that \( {\rho }_{M} \) is a \( \left( {*, \sim }\right) \) -good congruence on \( M \) when \( \rho \) is a \( \left( {*, \sim }\right) \) -good congruence on \( S \) . By Lemma 1... | Yes |
Theorem 1.1 Let \( S = I \times M \times \Lambda \) be a rectangular lc-monoid, and \( \rho \) be a \( \left( {*, \sim }\right) \) -good congruence on \( S \) . Then\n\n\[ \left( {i, a,\lambda }\right) \rho \left( {j, b,\mu }\right) \Leftrightarrow \left( {i,\lambda }\right) {\rho }_{E}\left( {j,\mu }\right), a{\rho }_... | Proof If \( \left( {i, a,\lambda }\right) \rho \left( {j, b,\mu }\right) \), then \( \left( {i,1,\lambda }\right) \rho \left( {j,1,\mu }\right) \), for \( \rho \) a \( \left( {*, \sim }\right) \) -good congruence on \( S \) . It follows that \( \left( {i,\lambda }\right) {\rho }_{E}\left( {j,\mu }\right) \) . Hence,\n\... | Yes |
Lemma 1.2 \( {}^{\left\lbrack 6\right\rbrack } \) Let \( B \) be a band. Then the following conditions are equivalent:\n\n(1) \( B \) is a \( {WLR} \) -band;\n\n(2) \( {D}_{e} = {L}_{e} \) or \( {D}_{e} = {R}_{e} \) for any \( e \in B \) ;\n\n(3) ef \( \mathcal{R}f \) e or ef \( \mathcal{L}f \) e for any \( e, f \in B ... | By the above results, we know that a semigroup \( B \) is a \( {WLR} \) -band if and only if each \( \mathcal{D} \) -class of \( B \) must be a left zero band or a right zero band. | No |
Lemma 2.1 \( {B}_{\alpha } \subseteq {B}_{L} \) or \( {B}_{\alpha } \subseteq {B}_{R} \) for any \( \alpha \in Y \) . | Proof If \( \left| {B}_{\alpha }\right| = 1 \), suppose that \( {B}_{\alpha } = \{ e\} \) . If efe \( = {ef} \) for all \( f \in B \), then \( {B}_{\alpha } \subseteq {B}_{L} \) ; similarly, if \( {efe} = {fe} \) for all \( f \in B \), then \( {B}_{\alpha } \subseteq {B}_{R} \) . If \( \left| {B}_{\alpha }\right| > 1 \... | Yes |
Lemma 2.2 \( {Y}_{L} \) and \( {Y}_{R} \) are subsemilattices of \( Y \), and \( Y = {Y}_{L} \cup {Y}_{R}, C\left( B\right) = \bigcup \left\{ {{B}_{\alpha } \mid \alpha \in }\right. \) \( \left. {{Y}_{L} \cap {Y}_{R}}\right\} \) | Proof For any \( e \in {B}_{\alpha }, f \in {B}_{\beta } \), where \( \alpha ,\beta \in {Y}_{L} \), then \( {ef} \in {B}_{\alpha \beta } \) . And for any \( g \in B \) , \( \left( {ef}\right) g\left( {ef}\right) = e\left( {fgf}\right) = \left( {ef}\right) g \) . By the definition of \( {LR} -regular bands, we can get t... | No |
Lemma 2.3 Let \( \alpha \in {Y}_{L},\beta \in {Y}_{R} \) (or \( \alpha \in {Y}_{R},\beta \in {Y}_{L} \) ) with \( \alpha \geq \beta \) . We have\n\n\[ \left| {B}_{d\left( {\alpha ,\beta }\right) }\right| = 1 \]\n\nfor any \( d\left( {\alpha ,\beta }\right) \in D\left( {\alpha ,\beta }\right) \) . | Proof Let \( \alpha \in {Y}_{L},\beta \in {Y}_{R} \) . For any \( e \in {B}_{\alpha } \), we have \( {exe} = {ex} \) for any \( x \in B \) . Then for any \( g, h \in {B}_{\beta } \), we have \( {ege} = {eg} \) , \( {ehe} = {eh} \) . If \( g{\rho }_{\alpha ,\beta }h \), then \( {eg} = {eh} \) . On the other hand, since ... | Yes |
Lemma 2.4 For any \( \alpha ,\beta \in Y,\alpha \in {Y}_{L},\beta \in {Y}_{R} \) (or \( \alpha \in {Y}_{R},\beta \in {Y}_{L} \) ) with \( \alpha \geq \beta \) and for any \( e \in {B}_{\alpha }, f \in {B}_{\beta } \) , \[ e{\theta }_{d\left( {\alpha ,\beta }\right) }^{f} = f\;\text{ and }\;{ef} = {fe} = f. \] | Proof From Lemma 2.3, we know that \( \left| {B}_{{d}^{\prime }\left( {\alpha ,\beta }\right) }\right| = 1 \) for any \( {d}^{\prime }\left( {\alpha ,\beta }\right) \in D\left( {\alpha ,\beta }\right) \) . On the other hand, by Conditions (A) and (C), there exists \( {B}_{d\left( {\alpha ,\beta }\right) } \) such that ... | Yes |
Theorem 3.1 A semigroup \( B \) is a \( {WLR} \) -regular band if and only if it is a refined semi-lattice of rectangular bands \( {B}_{\alpha } \), where \( \alpha \in Y, Y \) is a semilattice, and \( {B}_{\alpha } \) is a left zero band or a right zero band for each \( \alpha \in Y \) . | Proof The converse part is obvious by Lemmas 1.1 and 1.2. We only need to prove the direct part. Let \( e \in {B}_{\alpha } \) and \( f \in {B}_{\beta } \) . If \( {B}_{\alpha \beta } \) is a left zero band, by the multiplication of \( B \) as a refined semilattice of rectangular bands and Condition (A), \[ {efe} = \le... | Yes |
Lemma 4.3 An \( {LR} \)-regular band \( B \) is nice if and only if for any \( \alpha ,\beta \in Y,\alpha \in {Y}_{L},\beta \in {Y}_{R} \) (or \( \alpha \in {Y}_{R},\beta \in {Y}_{L} \)) with \( \alpha \geq \beta ,\left| {B}_{\beta }\right| = 1 \) . | Proof Let \( B \) be an \( {LR} \)-regular band. For any \( {\alpha }^{\prime },{\beta }^{\prime } \in Y \) with \( {\alpha }^{\prime } \in {Y}_{L},{\beta }^{\prime } \in {Y}_{R} \) and \( e \in {B}_{{\alpha }^{\prime }}, f \in {B}_{{\beta }^{\prime }} \). Then \( {\alpha }^{\prime }{\beta }^{\prime } \in {Y}_{L} \) or... | Yes |
Theorem 4.2 A semigroup \( B \) is an \( {LR} \) -normal band if and only if it is a strong semilattice of rectangular bands, denoted by \( B = \left\{ {Y;{B}_{\alpha },{\theta }_{\alpha ,\beta }}\right\} \), Condition (E) is satisfied and \( B \) is also nice. | Proof Let \( B \) be an \( {LR} \) -normal band. Then it is a strong semilattice of rectangular bands by Definition 1.4. On the other hand, since \( B \) is an \( {LR} \) -regular band, by Lemma 2.3, for any \( \alpha ,\beta \in Y,\alpha \in {Y}_{L},\beta \in {Y}_{R} \) or \( \alpha \in {Y}_{R},\beta \in {Y}_{L} \) wit... | Yes |
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