| {"id":"def:subgroup-of-semigroup","label":"subgroup-of-semigroup","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(S\\) be a semigroup. A subgroup of \\(S\\) is a subset \\(H \\subseteq S\\) equipped with: A carrier set \\(H \\subseteq S\\); Closure under multiplication: For all \\(a, b \\in H\\), we have \\(a \\cdot b \\in H\\); An identity element \\(e \\in H\\) such that: \\(e \\in H\\) (the identity is in the carrier); For all \\(x \\in H\\), \\(e \\cdot x = x\\) (left identity); For all \\(x \\in H\\), \\(x \\cdot e = x\\) (right identity); An inverse function \\(\\mathrm{inv} : S \\to S\\) such that: For all \\(x \\in H\\), \\(\\mathrm{inv}(x) \\in H\\) (closure under inverses); For all \\(x \\in H\\), \\(\\mathrm{inv}(x) \\cdot x = e\\) (left inverse); For all \\(x \\in H\\), \\(x \\cdot \\mathrm{inv}(x) = e\\) (right inverse). With these properties, the carrier \\(H\\) forms a group under the multiplication inherited from \\(S\\). Note that the identity element \\(e\\) is uniquely determined by the carrier set: if \\(e_1\\) and \\(e_2\\) are both identities for the same carrier \\(H\\), then \\(e_1 = e_1 \\cdot e_2 = e_2\\). Similarly, the inverse function is uniquely determined by the carrier and identity.","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.Subgroup","latex_link":"sect0006.html#def:subgroup-of-semigroup","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.Subgroup","lean_decl_name":"Semigroup.Subgroup","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"finalized","source_path":"MyProject/Substructures.lean","lean4_code":"/-- A Subgroup of a Semigroup `S` is a subset closed under multiplication and containing\nand identity element and an inverse function -/\nstructure Subgroup (S : Type*) [Semigroup S] extends Submonoid S where\n /-- The inverse function -/\n inv (x : S) : S\n /-- The inverse function is the identity if `x ∉ carrier`. This is to ensure\n injectivity with `Set` -/\n inv_not_mem {x : S} (hin : x ∉ carrier) : inv x = x\n /-- The inverses are in the carrier -/\n inv_mem {x : S} (hin : x ∈ carrier) : inv x ∈ carrier\n /-- The inverse function provides right inverses -/\n inv_mul {x : S} (hin : x ∈ carrier) : inv x * x = one\n /-- The inverse function provides left inverses -/\n mul_inv {x : S} (hin : x ∈ carrier) : x * inv x = one\n","line":178,"end_line":191,"proof_status":"formalized"} |
| {"id":"def:maximal-subgroup","label":"maximal-subgroup","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["def:subgroup-of-semigroup"],"nlp_text":"A subgroup \\(H\\) of a semigroup \\(S\\) is called maximal if it is not properly contained in any other subgroup of \\(S\\). That is, \\(H\\) is maximal if for every subgroup \\(K\\) of \\(S\\) with \\(H \\subseteq K\\), we have \\(H = K\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.Subgroup.isMaximal","latex_link":"sect0006.html#def:maximal-subgroup","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.Subgroup.isMaximal","lean_decl_name":"Semigroup.Subgroup.isMaximal","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"finalized","source_path":"MyProject/Substructures.lean","lean4_code":"/-- A maximal subgroup is a subgroup that is not properly contained in any other subgroup. -/\ndef isMaximal (H : Subgroup S) : Prop := ∀ (K : Subgroup S), H ≤ K → H = K\n","line":327,"end_line":328,"proof_status":"formalized"} |
| {"id":"lem:hclass-subgroup","label":"hclass-subgroup","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["def:maximal-subgroup","lem:location-theorem"],"nlp_text":"Let \\(S\\) be a semigroup and let \\(e \\in S\\) be an idempotent element. Then: The \\(\\mathcal{H}\\)-class \\([e]_{\\mathcal{H}}\\) is closed under multiplication. The element \\(e\\) acts as an identity on \\([e]_{\\mathcal{H}}\\): for all \\(x \\in [e]_{\\mathcal{H}}\\), \\(e \\cdot x = x\\) and \\(x \\cdot e = x\\). Every element \\(x \\in [e]_{\\mathcal{H}}\\) has an inverse \\(y \\in [e]_{\\mathcal{H}}\\) satisfying \\(x \\cdot y = e\\) and \\(y \\cdot x = e\\). The \\(\\mathcal{H}\\)-class \\([e]_{\\mathcal{H}}\\) forms a subgroup of \\(S\\) with identity \\(e\\). Every maximal subgroup of \\(S\\) is the \\(\\mathcal{H}\\)-class of some idempotent.","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.HEquiv.subgroup_of_idempotent","latex_link":"sect0007.html#lem:hclass-subgroup","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.HEquiv.subgroup_of_idempotent","lean_decl_name":"Semigroup.HEquiv.subgroup_of_idempotent","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"finalized","source_path":"./WorkInProgress/Location.lean","lean4_code":"noncomputable def HEquiv.subgroup_of_idempotent {e : S} (he : IsIdempotentElem e) : Subgroup S where\n carrier := ⟦e⟧𝓗\n mul_mem := HEquiv.mul_closed_of_idempotent he\n one := e\n one_mem := by simp\n one_mul {x : S} (hx : x 𝓗 e) := HEquiv.idempotent_mul he hx\n mul_one {x : S} (hx : x 𝓗 e) := HEquiv.mul_idempotent he hx\n inv (x : S) := by\n have hd : Decidable (x ∈ ⟦e⟧𝓗) := by exact Classical.propDecidable (x ∈ ⟦e⟧𝓗)\n exact (if hx : x ∈ ⟦e⟧𝓗\n then Exists.choose (HEquiv.exists_inverse_of_idempotent he hx)\n else x )\n inv_not_mem := by simp_all\n inv_mem := by\n simp_all\n intros x hx\n have h := Classical.choose_spec (HEquiv.exists_inverse_of_idempotent he hx)\n exact h.1\n inv_mul := by\n simp_all\n intros x hx\n have h := Classical.choose_spec (HEquiv.exists_inverse_of_idempotent he hx)\n exact h.2.2\n mul_inv := by\n simp_all\n intros x hx\n have h := Classical.choose_spec (HEquiv.exists_inverse_of_idempotent he hx)\n exact h.2.1\n\n","line":254,"end_line":282,"proof_status":"formalized"} |
| {"id":"lem:maximal-subgroups-isomorphic","label":"maximal-subgroups-isomorphic","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:hclass-subgroup"],"nlp_text":"Let \\(S\\) be a semigroup and let \\(H\\) and \\(K\\) be maximal subgroups of \\(S\\). If there exist \\(x \\in H\\) and \\(y \\in K\\) with \\(x \\mathcal{D} y\\), then \\(H\\) and \\(K\\) are isomorphic as groups.","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.DEquiv.maximal_subgroups_equiv","latex_link":"sect0007.html#lem:maximal-subgroups-isomorphic","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.DEquiv.maximal_subgroups_equiv","lean_decl_name":"Semigroup.DEquiv.maximal_subgroups_equiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/Location.lean","lean4_code":"/-- Two maximal subgroups of a 𝓓-class are isomorphic. -/\ntheorem DEquiv.maximal_subgroups_equiv {x y : S} {H K : Subgroup S}\n (hH : H.isMaximal) (hK : K.isMaximal) (hx : x ∈ H) (hy : y ∈ K) (hd : x 𝓓 y) :\n Nonempty (H ≃* K) := by\n obtain ⟨e₁, hi₁, h₁⟩ := HEquiv.hClass_of_subgroup hH\n obtain ⟨e₂, hi₂, h₂⟩ := HEquiv.hClass_of_subgroup hK\n simp [Subgroup.mem_def, h₁, h₂] at hx hy\n have he : e₁ 𝓓 e₂ := by\n apply HEquiv.to_dEquiv at hx\n apply HEquiv.to_dEquiv at hy\n apply DEquiv.trans hx.symm\n apply DEquiv.trans hd hy\n have heq₁ : H = HEquiv.subgroup_of_idempotent hi₁ := by\n apply SetLike.ext\n simp [Subgroup.mem_def, h₁]\n have heq₂ : K = HEquiv.subgroup_of_idempotent hi₂ := by\n apply SetLike.ext\n simp [Subgroup.mem_def, h₂]\n rw [heq₁, heq₂]\n exact DEquiv.hClass_equiv hi₁ hi₂ he","line":505,"end_line":524,"proof_status":"formalized"} |
| {"id":"lem:exists-pow-sandwich","label":"exists-pow-sandwich","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:exists-unique-idempotent-pow"],"nlp_text":"In a finite monoid \\(M\\), if an element \\(a\\) satisfies the property \\(a = xay\\) for some \\(x, y \\in M\\), then there exist non-zero powers of \\(x\\) and \\(y\\), say \\(n_1\\) and \\(n_2\\), such that \\(x^{n_1}a = a\\) and \\(ay^{n_2} = a\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Monoid.exists_pow_sandwich_eq_self","latex_link":"sect0001.html#lem:exists-pow-sandwich","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Monoid.exists_pow_sandwich_eq_self","lean_decl_name":"Monoid.exists_pow_sandwich_eq_self","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"finalized","source_path":"MyProject/Idempotent.lean","lean4_code":"/-- In finite monoids, elements satisfying a sandwich property have powers that left-cancel\nand right-cancel the sandwich factors. -/\ntheorem exists_pow_sandwich_eq_self [Finite M] {x a y : M} (h : x * a * y = a) :\n ∃ n₁ n₂ : ℕ, n₁ ≠ 0 ∧ n₂ ≠ 0 ∧ x ^ n₁ * a = a ∧ a * y ^ n₂ = a := by\n have loop : ∀ k : ℕ, x ^ k * a * y ^ k = a := by\n intro k; induction k with\n | zero => simp\n | succ n ih =>\n rw [pow_succ, pow_succ']\n rw [← mul_assoc, mul_assoc _ a, mul_assoc _ x, ← mul_assoc x a y, h, ih]\n have ⟨n₁, ⟨hn₁, hneq₁⟩⟩ := Monoid.exists_idempotent_pow x\n have ⟨n₂, ⟨hn₂, hneq₂⟩⟩ := Monoid.exists_idempotent_pow y\n use n₁, n₂\n constructor\n · exact hneq₁\n constructor\n · exact hneq₂\n constructor\n · rw [← (loop n₁), ← mul_assoc, ← mul_assoc, hn₁]\n · rw [← (loop n₂), mul_assoc, hn₂]\n","line":122,"end_line":141,"proof_status":"formalized"} |
| {"id":"lem:d-j-theorem","label":"d-j-theorem","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:exists-pow-sandwich","def:greens-relations"],"nlp_text":"In a finite semigroup, the \\(\\mathcal{D}\\) and \\(\\mathcal{J}\\) relations are equivalent. That is, for any two elements \\(x, y \\in S\\), \\(x \\mathcal{D} y\\) if and only if \\(x \\mathcal{J} y\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.JEquiv.to_dEquiv","latex_link":"sect0004.html#lem:d-j-theorem","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.JEquiv.to_dEquiv","lean_decl_name":"Semigroup.JEquiv.to_dEquiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/Finite.lean","lean4_code":"/-- In finite semigroups, 𝓙-equivalence implies 𝓓-equivalence. -/\n@[simp] lemma JEquiv.to_dEquiv (hj : x 𝓙 y) : x 𝓓 y := by\n have hj₁ := hj\n obtain ⟨⟨s, t, ha⟩, ⟨u, v, hb⟩⟩ := hj₁\n have hab : s * u * x * (v * t) = ↑x := by\n have hrw : s * u * ↑x * (v * t) = s * (u * ↑x * v * t) := by simp [mul_assoc]\n rw [hrw, hb, ← mul_assoc, ha]\n obtain ⟨k, ⟨l, ⟨hkne, hlne, heq₁, heq₂⟩⟩⟩ := Monoid.exists_pow_sandwich_eq_self hab\n cases v with\n | one =>\n use x\n simp at ⊢ hb heq₂ hab\n constructor -- we prove `x 𝓛 y`\n · use (s * u)^(k-1) * s\n have hk : k - 1 + 1 = k := by exact Nat.succ_pred_eq_of_ne_zero hkne\n simp_rw [← hb, ← mul_assoc, mul_assoc _ s u, ← _root_.pow_succ, hk]\n simp [heq₁]\n · use u\n | coe v =>\n use x * v\n simp [REquiv, LEquiv]\n constructor\n · use t * (v * t) ^ (l - 1) -- `x ≤𝓡 x * v`\n rw [WithOne.coe_mul, ← mul_assoc, mul_assoc ↑x ↑v t]\n rw [mul_assoc ↑x (↑v * t), ← pow_succ']\n have hl : l - 1 + 1 = l := by exact Nat.succ_pred_eq_of_ne_zero hlne\n rw [hl, heq₂]\n · constructor\n · use (s * u)^(k-1) * s -- `x * v ≤𝓛 y`\n rw [← hb]\n have hk : k - 1 + 1 = k := by exact Nat.succ_pred_eq_of_ne_zero hkne\n conv => lhs; rw [← mul_assoc, ← mul_assoc, mul_assoc _ s u]\n rw [WithOne.coe_mul, ← _root_.pow_succ, hk, heq₁]\n · use u -- `y ≤𝓛 x * v`\n simp [← mul_assoc, hb]","line":66,"end_line":100,"proof_status":"formalized"} |
| {"id":"lem:j-strengthening","label":"j-strengthening","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:d-j-theorem"],"nlp_text":"In a finite semigroup, if two elements are \\(\\mathcal{J}\\)-equivalent, then a one-sided preorder implies the corresponding one-sided equivalence. Specifically, if \\(x \\mathcal{J} y\\) and \\(x \\leq _{\\mathcal{R}} y\\), then \\(x \\mathcal{R} y\\). Similarly, if \\(x \\mathcal{J} y\\) and \\(x \\leq _{\\mathcal{L}} y\\), then \\(x \\mathcal{L} y\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.REquiv.of_rPreorder_and_jEquiv","latex_link":"sect0004.html#lem:j-strengthening","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.REquiv.of_rPreorder_and_jEquiv","lean_decl_name":"Semigroup.REquiv.of_rPreorder_and_jEquiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/Finite.lean","lean4_code":"/-- In finite semigroups, 𝓙-equivalence strengthens the 𝓡-preorder to 𝓡-equivalence. -/\ntheorem REquiv.of_rPreorder_and_jEquiv (hr : x ≤𝓡 y) (hj : x 𝓙 y) : x 𝓡 y := by\n obtain ⟨z, hz⟩ := hr\n cases z with\n | one =>\n have heq : x = y := by simp_all\n subst x; simp\n | coe z =>\n have heq : y * z = x := by\n rw [← WithOne.coe_inj, WithOne.coe_mul]\n exact hz\n subst x\n symm\n apply REquiv.of_jEquiv_mul_right hj.symm","line":137,"end_line":150,"proof_status":"formalized"} |
| {"id":"lem:h-of-sandwich","label":"h-of-sandwich","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:j-strengthening"],"nlp_text":"In a finite semigroup, if an element \\(x\\) can be written as \\(x = uxv\\) for some \\(u, v \\in S\\), then \\(x\\) is \\(\\mathcal{H}\\)-equivalent to both \\(ux\\) and \\(xv\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.HEquiv.of_eq_sandwich","latex_link":"sect0004.html#lem:h-of-sandwich","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.HEquiv.of_eq_sandwich","lean_decl_name":"Semigroup.HEquiv.of_eq_sandwich","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/Finite.lean","lean4_code":"/-- In finite semigroups, an element sandwiched between two factors is 𝓗-related to its\nleft and right partial products. -/\ntheorem HEquiv.of_eq_sandwich (h : u * x * v = x) : x 𝓗 u * x ∧ x 𝓗 x * v := by\n simp [HEquiv.iff_rEquiv_and_lEquiv]\n constructor <;> constructor\n · apply REquiv.of_rPreorder_and_jEquiv\n · use v\n simpa [← WithOne.coe_mul]\n · simp [JEquiv]\n use 1, ↑v\n simpa [← WithOne.coe_mul]\n · apply LEquiv.of_jEquiv_mul_left\n simp [JEquiv]\n use 1, v\n simpa [← WithOne.coe_mul]\n · apply REquiv.of_jEquiv_mul_right\n simp [JEquiv]\n use u, 1\n simpa [← WithOne.coe_mul, ← mul_assoc]\n · apply LEquiv.of_lPreorder_and_jEquiv\n · use u\n simpa [← WithOne.coe_mul, ← mul_assoc]\n · simp [JEquiv]\n use u, 1\n simpa [← WithOne.coe_mul, ← mul_assoc]","line":168,"end_line":192,"proof_status":"formalized"} |
| {"id":"lem:location-theorem","label":"location-theorem","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:j-strengthening","lem:greens-lemma","lem:le-idempotent"],"nlp_text":"Let \\(S\\) be a semigroup and let \\(x, y \\in S\\). The following conditions are equivalent: \\(x \\cdot y \\in [x]_{\\mathcal{R}} \\cap [y]_{\\mathcal{L}}\\) (i.e., \\(x \\cdot y \\mathcal{R} x\\) and \\(x \\cdot y \\mathcal{L} y\\)). There exists an idempotent \\(e \\in [y]_{\\mathcal{R}} \\cap [x]_{\\mathcal{L}}\\). Moreover, in a finite semigroup, these conditions are equivalent to: \\(x \\mathcal{D} y\\) and \\(x \\cdot y \\mathcal{D} x\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.DEquiv.mul_in_inter_iff_equiv","latex_link":"sect0007.html#lem:location-theorem","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.DEquiv.mul_in_inter_iff_equiv","lean_decl_name":"Semigroup.DEquiv.mul_in_inter_iff_equiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/Location.lean","lean4_code":"/-- In Finite semigroups, `x * y` is in the intersection of the 𝓡-class of `x` and the 𝓛-class\nof `y` iff `x`, `y`, and `x * y` are 𝓓-Equivalent. -/\ntheorem DEquiv.mul_in_inter_iff_equiv [Finite S] : x * y ∈ ⟦x⟧𝓡 ∩ ⟦y⟧𝓛 ↔ x 𝓓 y ∧ x * y 𝓓 x := by\n simp_all\n constructor\n · rintro ⟨hr, hl⟩\n constructor\n · use x * y\n exact ⟨hr.symm, hl⟩\n · exact JEquiv.to_dEquiv <| REquiv.to_jEquiv hr\n · rintro ⟨hj₁, hj₂⟩\n apply DEquiv.to_jEquiv at hj₁\n apply DEquiv.to_jEquiv at hj₂\n constructor\n · refine REquiv.of_rPreorder_and_jEquiv ?_ hj₂\n simp\n · refine LEquiv.of_lPreorder_and_jEquiv ?_ ?_\n · simp\n · apply JEquiv.trans hj₂ hj₁","line":84,"end_line":102,"proof_status":"formalized"} |
| {"id":"lem:greens-relations-mul-compat","label":"greens-relations-mul-compat","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["def:greens-relations"],"nlp_text":"The \\(\\mathcal{R}\\) and \\(\\mathcal{L}\\) relations exhibit compatibility with semigroup multiplication on one side. Specifically, the \\(\\mathcal{R}\\)-preorder is compatible with left multiplication, and the \\(\\mathcal{L}\\)-preorder is compatible with right multiplication. If \\(x \\leq _{\\mathcal{R}} y\\), then for any \\(z \\in S\\), we have \\(zx \\leq _{\\mathcal{R}} zy\\). This property extends to the equivalence relation \\(\\mathcal{R}\\). If \\(x \\mathcal{R} y\\), then \\(zx \\mathcal{R} zy\\). A similar argument holds for the \\(\\mathcal{L}\\)-preorder and \\(\\mathcal{L}\\)-equivalence, which are compatible with right multiplication. If \\(x \\leq _{\\mathcal{L}} y\\), then \\(xz \\leq _{\\mathcal{L}} yz\\) for any \\(z \\in S\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.RPreorder.lmult_compat","latex_link":"sect0002.html#lem:greens-relations-mul-compat","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.RPreorder.lmult_compat","lean_decl_name":"Semigroup.RPreorder.lmult_compat","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/Defs.lean","lean4_code":"/-- The 𝓡-preorder is compatible with left multiplication. -/\n@[simp] lemma RPreorder.lmult_compat {x y : S} (h : x ≤𝓡 y) (z : S) : z * x ≤𝓡 z * y := by\n obtain ⟨u, hu⟩ := h; use u\n simp [mul_assoc, hu]","line":394,"end_line":397,"proof_status":"formalized"} |
| {"id":"lem:greens-lemma","label":"greens-lemma","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:greens-relations-mul-compat"],"nlp_text":"Let \\(S\\) be a semigroup and suppose \\(x \\mathcal{R} y\\) with witnesses \\(u, v \\in S\\) such that \\(x \\cdot v = y\\) and \\(y \\cdot u = x\\). Then: The map \\(\\rho _v : w \\mapsto w \\cdot v\\) is a bijection from the \\(\\mathcal{L}\\)-class of \\(x\\) to the \\(\\mathcal{L}\\)-class of \\(y\\). The map \\(\\rho _u : w \\mapsto w \\cdot u\\) is the inverse of \\(\\rho _v\\). These bijections preserve \\(\\mathcal{H}\\)-classes: for any \\(a, b\\) in the \\(\\mathcal{L}\\)-class of \\(x\\), we have \\(a \\mathcal{H} b\\) if and only if \\(a \\cdot v \\mathcal{H} b \\cdot v\\). Dually, if \\(x \\mathcal{L} y\\) with witnesses \\(u, v\\) such that \\(v \\cdot x = y\\) and \\(u \\cdot y = x\\), then left translation \\(\\lambda _v : w \\mapsto v \\cdot w\\) is a bijection from the \\(\\mathcal{R}\\)-class of \\(x\\) to the \\(\\mathcal{R}\\)-class of \\(y\\), with inverse \\(\\lambda _u\\), and these bijections preserve \\(\\mathcal{H}\\)-classes.","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.REquiv.bijOn_lClass","latex_link":"sect0005.html#lem:greens-lemma","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.REquiv.bijOn_lClass","lean_decl_name":"Semigroup.REquiv.bijOn_lClass","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/GreensLemma.lean","lean4_code":"/-- If `x 𝓡 y` such that `x * v = y`, then the map `w ↦ w * v` is a bijection from\nthe 𝓛-class of `x` to that of `y`. -/\ntheorem REquiv.bijOn_lClass (hr : x 𝓡 y) (hv : x * v = y) :\n Set.BijOn (fun w ↦ w * v) ⟦x⟧𝓛 ⟦y⟧𝓛 := by\n refine Set.BijOn.mk ?_ ?_ ?_\n · apply RPreorder.mapsTo_lClass hv\n · apply hr.injOn_lClass hv\n · apply hr.surjOn_lClass hv","line":131,"end_line":138,"proof_status":"formalized"} |
| {"id":"lem:r-l-comm","label":"r-l-comm","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:greens-relations-mul-compat"],"nlp_text":"The composition of the relations \\(\\mathcal{R}\\) and \\(\\mathcal{L}\\) is commutative. That is, for any \\(x, y \\in S\\), there exists a \\(z\\) such that \\(x \\mathcal{R} z\\) and \\(z \\mathcal{L} y \\iff \\) there exists a \\(w\\) such that \\(x \\mathcal{L} w\\) and \\(w \\mathcal{R} y\\). This can be written as \\(\\mathcal{R} \\circ \\mathcal{L} = \\mathcal{L} \\circ \\mathcal{R}\\). This property is crucial for proving that the relation \\(\\mathcal{D} = \\mathcal{R} \\circ \\mathcal{L}\\) is symmetric, and therefore an equivalence relation.","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.rEquiv_lEquiv_comm","latex_link":"sect0002.html#lem:r-l-comm","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.rEquiv_lEquiv_comm","lean_decl_name":"Semigroup.rEquiv_lEquiv_comm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"finalized","source_path":"MyProject/Green/Defs.lean","lean4_code":"/-- The 𝓡 and 𝓛 relations commute under composition. -/\ntheorem rEquiv_lEquiv_comm (x y : S) : (∃ z, x 𝓡 z ∧ z 𝓛 y) ↔ (∃ z, x 𝓛 z ∧ z 𝓡 y) := by\n constructor\n · rintro ⟨z, ⟨hr, hl⟩⟩\n have hr₁ := hr\n have hl₁ := hl\n rcases hr₁ with ⟨⟨w₁, hw₁⟩, ⟨v₁, hv₁⟩⟩\n rcases hl₁ with ⟨⟨w₂, hw₂⟩, ⟨v₂, hv₂⟩⟩\n -- We have `x = w₂ * y * w₁` and `y = v₂ * z * v₁`\n -- We want to use `v₂ * z * w₁`, but we need to destruct on `v₂, w₁ : WithOne S`\n cases hv₂one : v₂ with\n | one =>\n subst v₂ -- trivial case where `v₂ = 1`\n simp at hv₂\n subst z -- `z = y`\n use x; simp [hr]\n | coe v₂ =>\n subst hv₂one\n cases hw₁one : w₁ with\n | one =>\n subst w₁ -- trivial case where `w₁ = 1` and `z = x`\n simp at hw₁\n subst hw₁\n use y; simp [hl]\n | coe w₁ => -- Non-trivial case, use `v₂ * z * w₁`\n subst hw₁one\n use v₂ * z * w₁\n constructor\n <;> constructor\n · use w₂ -- `x ≤𝓛 v₂ * z * w₁`\n simp [← mul_assoc]\n rw [mul_assoc w₂, hv₂, hw₂, ← hw₁]\n · use v₂ -- `v₂ * z * w₁ ≤𝓛 x`\n simp [mul_assoc]\n rw [← hw₁]\n · use w₁ -- `v₂ * z * w₁ ≤𝓡 y`\n simp [hv₂]\n · use v₁ -- `y ≤𝓡 v₂ * z * w₁`\n simp [mul_assoc v₂]\n rw [hw₁, ← hv₂, ← hv₁]\n simp [mul_assoc]\n · rintro ⟨z, ⟨hl, hr⟩⟩\n have hr₁ := hr\n have hl₁ := hl\n rcases hr₁ with ⟨⟨w₁, hw₁⟩, ⟨v₁, hv₁⟩⟩\n rcases hl₁ with ⟨⟨w₂, hw₂⟩, ⟨v₂, hv₂⟩⟩\n -- We have `y = v₂ * x * v₁` and `x = w₂ * y * w₁`\n -- We want to use `w₂ * z * v₁`, but we need to destruct on `w₂, v₁ : WithOne S`\n cases hw₂one : w₂ with\n | one =>\n subst w₂ -- trivial case where `w₂ = 1`\n simp at hw₂\n subst z -- `z = x`\n use y; simp [hr]\n | coe w₂ =>\n subst hw₂one\n cases hv₁one : v₁ with\n | one =>\n subst v₁ -- trivial case where `v₁ = 1` and `z = y`\n simp at hv₁\n subst hv₁\n use x; simp [hl]\n | coe v₁ => -- Non-trivial case, use `w₂ * z * v₁`\n subst hv₁one\n use w₂ * z * v₁\n constructor\n <;> constructor\n · use w₁; simp -- `x ≤𝓡 w₂ * z * v₁`\n nth_rw 1 [← hw₂]\n conv => lhs; lhs; rw [mul_assoc, hv₁]\n rw [mul_assoc, hw₁]\n · use v₁ -- `w₂ * z * v₁ ≤𝓡 x`\n simp [hw₂]\n · use w₂ -- `w₂ * z * v₁ ≤𝓛 y`\n simp [← hv₁, ← mul_assoc]\n · use v₂ -- `y ≤𝓛 w₂ * z * v₁`\n simp [hw₂, ← mul_assoc, hv₂, hv₁]\n","line":413,"end_line":489,"proof_status":"formalized"} |
| {"id":"lem:d-equiv-closed","label":"d-equiv-closed","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:r-l-comm"],"nlp_text":"The \\(\\mathcal{D}\\) relation is closed under composition with \\(\\mathcal{R}\\) and \\(\\mathcal{L}\\). If \\(x \\mathcal{D} y\\) and \\(y \\mathcal{L} z\\), then \\(x \\mathcal{D} z\\). Similarly, if \\(x \\mathcal{D} y\\) and \\(y \\mathcal{R} z\\), then \\(x \\mathcal{D} z\\). This property is used to prove the transitivity of \\(\\mathcal{D}\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.DEquiv.closed_under_lEquiv","latex_link":"sect0002.html#lem:d-equiv-closed","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.DEquiv.closed_under_lEquiv","lean_decl_name":"Semigroup.DEquiv.closed_under_lEquiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"finalized","source_path":"MyProject/Green/Defs.lean","lean4_code":"/-- The 𝓓 relation is closed under composition with 𝓛. -/\nlemma closed_under_lEquiv {x y z : S} (hd : x 𝓓 y) (hl : y 𝓛 z) : x 𝓓 z := by\n rcases hd with ⟨w, ⟨hw1, hw2⟩⟩\n use w\n simp_all\n apply LEquiv.trans hw2 hl\n","line":511,"end_line":516,"proof_status":"formalized"} |
| {"id":"def:greens-relations","label":"greens-relations","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"In a semigroup \\(S\\), Green’s relations are a set of five equivalence relations that characterize the elements of \\(S\\) in terms of the principal ideals they generate. These relations are fundamental to the study of semigroups. First, we define four preorder relations: \\(\\leq _{\\mathcal{R}}\\), \\(\\leq _{\\mathcal{L}}\\), \\(\\leq _{\\mathcal{J}}\\), and \\(\\leq _{\\mathcal{H}}\\). Let \\(S^1\\) be the monoid obtained by adjoining an identity element to \\(S\\) if it does not already have one. For any two elements \\(x, y \\in S\\): \\(x \\leq _{\\mathcal{R}} y\\) iff the principal right ideal generated by \\(x\\) is a subset of the principal right ideal generated by \\(y\\) (\\(xS^1 \\subseteq yS^1\\)). In Lean, we use the equivalent definition that there must exist some \\(z \\in S^1\\) such that \\(x = yz\\). \\(x \\leq _{\\mathcal{L}} y\\) iff the principal left ideal generated by \\(x\\) is a subset of the principal left ideal generated by \\(y\\) (\\(S^1x \\subseteq S^1y\\)). In Lean, we use the equivalent definition that there must exist some \\(z \\in S^1\\) such that \\(x = zy\\). \\(x \\leq _{\\mathcal{J}} y\\) iff the principal two-sided ideal generated by \\(x\\) is a subset of the principal two-sided ideal generated by \\(y\\) (\\(S^1xS^1 \\subseteq S^1yS^1\\)). In Lean, we use the equivalent definition that there must exist some \\(u, v \\in S^1\\) such that \\(x = uyv\\). \\(x \\leq _{\\mathcal{H}} y \\iff x \\leq _{\\mathcal{R}} y\\) and \\(x \\leq _{\\mathcal{L}} y\\). These four relations are preorders (they are reflexive and transitive). From these preorders, we define the equivalence relations \\(\\mathcal{R}\\), \\(\\mathcal{L}\\), \\(\\mathcal{J}\\), and \\(\\mathcal{H}\\) as the symmetric closures of their corresponding preorders. For example, \\(x \\mathcal{R} y \\iff x \\leq _{\\mathcal{R}} y\\) and \\(y \\leq _{\\mathcal{R}} x\\). The equivalence classes are denoted by \\([x]_{\\mathcal{R}}\\), \\([x]_{\\mathcal{L}}\\), etc. Finally, the \\(\\mathcal{D}\\) relation is defined by composing \\(\\mathcal{R}\\) and \\(\\mathcal{L}\\): \\(x \\mathcal{D} y\\) if there exists an element \\(z \\in S\\) such that \\(x \\mathcal{R} z\\) and \\(z \\mathcal{L} y\\). It can be shown that \\(\\mathcal{D}\\) is an equivalence relation and that it can also be defined by composing \\(\\mathcal{L}\\) and \\(\\mathcal{R}\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.RPreorder","latex_link":"sect0002.html#def:greens-relations","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.RPreorder","lean_decl_name":"Semigroup.RPreorder","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"finalized","source_path":"./MyProject/Green/GreensLemma.lean","lean4_code":"lemma RPreorder.mapsTo_lClass (hv : x * v = y) :\n Set.MapsTo (fun w ↦ w * v) ⟦x⟧𝓛 ⟦y⟧𝓛 := by\n simp [Set.MapsTo]\n intros z hz\n rw [← hv]\n apply LEquiv.rmult_compat\n exact hz\n\n/-- If `x 𝓡 y` such that `x * v = y` then the map `w ↦ w * v` is injective on the\n𝓛-class of `x`. -/\n","line":65,"end_line":74,"proof_status":"formalized"} |
| {"id":"lem:le-idempotent","label":"le-idempotent","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["def:greens-relations"],"nlp_text":"Let \\(e\\) be an idempotent element in a semigroup \\(S\\). An element \\(x \\in S\\) is \\(\\mathcal{R}\\)-below \\(e\\) if and only if \\(x = ex\\). Similarly, \\(x\\) is \\(\\mathcal{L}\\)-below \\(e\\) if and only if \\(x = xe\\). It follows that \\(x\\) is \\(\\mathcal{H}\\)-below \\(e\\) if and only if both conditions hold, i.e., \\(x = ex\\) and \\(x = xe\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.RPreorder.le_idempotent","latex_link":"sect0003.html#lem:le-idempotent","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.RPreorder.le_idempotent","lean_decl_name":"Semigroup.RPreorder.le_idempotent","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/Basic.lean","lean4_code":"/-- An element `x` is 𝓡-below an idempotent `e` if and only if `x = e * x`. -/\ntheorem RPreorder.le_idempotent {e : S} (h : IsIdempotentElem e) (x : S) :\n (x ≤𝓡 e) ↔ (e * x = x) := by\n constructor\n · rintro ⟨u, hru⟩\n unfold IsIdempotentElem at h\n rw [← WithOne.coe_inj, WithOne.coe_mul] at h ⊢\n rw [← hru, ← mul_assoc, h]\n · intro hl; use x\n rw [← WithOne.coe_inj] at hl\n exact hl","line":65,"end_line":75,"proof_status":"formalized"} |
| {"id":"lem:greens-relations-hom-pres","label":"greens-relations-hom-pres","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["def:greens-relations"],"nlp_text":"Green’s relations are preserved under semigroup morphisms. Let \\(f: S \\to T\\) be a semigroup morphism. If two elements \\(x, y \\in S\\) are related by any of Green’s preorders or equivalence relations, then their images \\(f(x), f(y)\\) are related by the same relation in \\(T\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.RPreorder.hom_pres","latex_link":"sect0003.html#lem:greens-relations-hom-pres","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.RPreorder.hom_pres","lean_decl_name":"Semigroup.RPreorder.hom_pres","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/Basic.lean","lean4_code":"/-- The 𝓡-preorder is preserved by semigroup morphisms. -/\ntheorem RPreorder.hom_pres (f : F) (x y : S) (h : x ≤𝓡 y) : f x ≤𝓡 f y := by\n obtain ⟨z, hz⟩ := h\n cases z with\n | one => simp_all\n | coe z =>\n have heq : x = y * z := by\n rw [← WithOne.coe_inj, WithOne.coe_mul]\n exact hz.symm\n rw [← WithOne.coe_mul, WithOne.coe_inj] at hz\n subst x\n simp","line":115,"end_line":126,"proof_status":"formalized"} |
| {"id":"lem:exists-unique-idempotent-pow","label":"exists-unique-idempotent-pow","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"In a finite semigroup, for any element \\(x\\), there exists a positive integer \\(m\\) such that \\(x^m\\) is an idempotent. This idempotent power is unique.","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.exists_repeating_pow","latex_link":"sect0001.html#lem:exists-unique-idempotent-pow","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.exists_repeating_pow","lean_decl_name":"Semigroup.exists_repeating_pow","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"finalized","source_path":"MyProject/Idempotent.lean","lean4_code":"/-- In a finite semigroup, powers of any element eventually repeat. -/\nlemma exists_repeating_pow [Finite S] (x : S) : ∃ (m n : ℕ+), x ^ m * x ^ n = x ^ m := by\n obtain ⟨o, p, hop, heq⟩ : ∃ o p : ℕ+, o ≠ p ∧ x ^ o = x ^ p := by\n apply Finite.exists_ne_map_eq_of_infinite\n simp_all only [ne_eq, pow_add]; pnat_to_nat\n rcases (Nat.lt_or_gt_of_ne hop) with (o_lt_p | p_lt_o)\n · use o, p - o; simp_all\n · use p, o - p; simp_all\n","line":51,"end_line":58,"proof_status":"formalized"} |
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