Title: Theory / Unimportant / CS / Fundamental / Automata / Finite / Pin{ / A Automata and semigroups / 05 Green's relation and local theory
Tags: fromMath automaticImplies mathToDef toBinaryRelation multipartite complex hasSeenConjDef counterExample typToBinary defToAlgebra equivalentHand fromAlgorithm hasNotNewEffectReturnsMeaning oldAlgoNewName récupéré setMultiplication
On every note of this deck, unless a note says otherwise:
Chapter: V

Page: 101
Definition: \(e\le_{\mathcal H}f\)
Chapter: V
Section: 1
Variables: \(S\) be a semigroup.
\(e,f\in E(S)\)
Notation: \(e\le f\)
Kind: Prop, Cor
Index: 4, 5
Definition2: \(ef=e=fe\)
Extra: 1 and 2 are equivalent even if \(f\) is non idempotent.
2 and 3 follow from simplification lemma
Definition3: \(fef=e\)

Name: The natural order
Definition: The restriction of \(\le_{\mathcal H}\) to \(E(S)\).
Page: 101
Chapter: V
Section: 1
Variables: \(S\) be a semigroup.
Notation: \(\le\)
Index: 4, 5
Kind: Prop, Cor

Smaller0: \(S\) is finite
Greater0: \(s\mathcal L sx\)
Page: 103
Section: 1
Variables: \(S\) a semigroup, \(s,x\in S\).
Index: 9 (1)
Kind: Theo
Smaller1: \(s\le_{\mathcal J}sx\)
Construction: \(uxsv=s\) for some \(uv\in S^1\)
\((ux)^{n}sv^{n}=(ux)^{n-1}sv^{n-1}=s\) by induction
for \(n\) such that \(v^{n}\) idempotent,
\(s=(ux)^nsv^n=(ux)^nsv^nv^n=sv^n\)
\(((ux)^{n-1}u)xs=(ux)^ns=(ux)^nsv^n=s\) and thus \(s\mathcal Lxs\)

Smaller0: \(S\) is finite
Greater0: \(s\mathcal R sx\)
Page: 103
Section: 1
Variables: \(S\) a semigroup
\(s,x\in S\)
Index: 9(2)
Kind: Theo
Smaller1: \(s\le_{\mathcal J}sx\)
Construction: \(usxv=s\) for some \(uv\in S^1\)
\(u^{n}s(xv)^{n}=u^{n-1}s(xv)^{n-1}=s\) by induction
for \(n\) such that \(u^{n}\) idempotent,
\(s=u^ns(xv)^n=u^nu^ns(xv)^n=u^ns\)
\(sx(v(xv)^{n-1})=s(xv)^n=u^ns(xv)^n=s\) and thus \(s\mathcal Rsx\)

Smaller0: \(S\) is finite
Greater0: \(s\mathcal Lt\)
Page: 103
Section: 1
Variables: \(S\) a semigroup,
\(s,t\in S\).
Index: 9(4)
Kind: Theo
Hide middle: yes
Smaller1: \(s\le_{\mathcal L}t\)
Construction: \(s=ut\) for some \(u\in S^1\)
\(ut=s\mathcal Jt\).
By another proposition, \(S\) finite, \(t\le_{\mathcal J}ut\) implies \(t\mathcal L ut=s\)
Smaller2: \(s\mathcal Jt\)

Smaller0: \(S\) is finite
Greater0: \(s\mathcal Rt\)
Page: 103
Section: 1
Variables: \(S\) a semigroup, \(s,t\in S\).
Index: 9(3)
Kind: Theo
Hide middle: yes
Smaller1: \(s\le_{\mathcal R}t\)
Construction: \(s=tu\) for some \(u\in S^1\)
\(tu=s\mathcal Jt\).
By another proposition, \(S\) finite, \(t\le_{\mathcal J}tu\) implies \(t\mathcal R tu=s\)
Smaller2: \(s\mathcal Jt\)

Question: \(\mathcal H\)-classes are
-closed under right multiplication by
Answer: an element \(p\) such that there exists \(q\) such that \(s=tp\) and \(t=sq\)
Page: 103
Chapter: V
Section: 1
Variables: \(S\) a semigroup,
\(s,t\in S\),
\(p,q\in S^1\).
Index: 10
Kind: Prop
Name: Green's Lemma

Page: 103
Implementation: \(x\to xp\)
Section: 1
Chapter: V
Variables: \(S\) a semigroup,
\(s,t\in S\),
\(p,q\in S^1\).
Name: Green's Lemma
Index: 10
Kind: Prop
Typ: a bijection from \(L(t)\) onto \(L(s)\)
Assuming: \(s=tp\) and \(t=sq\)

Smaller0: \(st\in R(s)\cap L(t)\)
Greater0: \(\overline sst=t\) for some inverse \(\overline s\) of \(s\)
Page: 104, 105
Section: 1
Variables: \(S\) a semigroup,
\(D\) be a \(\mathcal D\)-class of a semigroup, and
\(s,t\in D\).
Index: 11
Kind: Theo
Smaller to greater: iff
Greater1: \(st\overline t=s\) for some inverse \(\overline t\) of \(t\)
Smallest: \(R(t)\cap L(s)\) contains an idempotent
Greatest: \(st\in D\)

Smaller0: \(H\) is a group
Greater0: \(H\) contains an idempotent
Page: 105
Section: 1
Variables: \(S\) a semigroup
\(H\) be a \(\mathcal H\)-class of \(S\).
Index: 13
Kind: Prop
Smaller to greater: iff
Greatest: there exists \(s,t\in H\) such that \(st\in H\)

Smaller0: \(S\) is finite
Greater0: \(H(t)\) is a group
Page: 106
Section: 1
Variables: \(S\) be a semigroup
\(s,t\in S\).
Index: 15
Kind: Cor
Smaller1: \(s\mathcal J t\)
Smaller2: (\(st\mathcal L s\) or \(ts\mathcal R s\))

Smaller0: \(S\) is finite
Greater0: \(t\) is idempotent
Page: 105
Section: 1
Variables: \(S\) be a finite semigroup and \(t\in S\).
Index: 15
Kind: Cor
Smaller1: there exists \(s\in S\), \(\mathcal J\)-related to \(t\) and such that \(st=s\)

Name: An inverse of \(s\in S\)
Definition: An element \(t\) such that both elements are weak inverse of the other one
Page: 106
Chapter: V
Section: 2.1
Variables: \(S\) a semigroup,
\(s\in S\).
Notation: \(\overline s\)

Name: A weak inverse of \(s\in S\)
Definition: An element \(t\) such that \(tst=t\).
Page: 103
Chapter: V
Section: 2.1
Variables: \(S\) a semigroup
\(s\in S\).
Context: Semigroup
Notation: \(\overline s\)

Definition: The set of weak inverses of \(s\).
Page: 106
Chapter: 5
Section: 2.1
Variables: \(S\) a semigroup
\(s\in S\)
Notation: \(W(s)\)

Definition: The set of inverse of \(s\).
Page: 106
Chapter: V
Section: 2.1
Variables: \(S\) a semigroup, \(s\in S\).
Notation: \(V(s)\)

Smaller0: \(t\in V(s)\)
Greater0: \(\mathcal D\)
Page: 107
Section: 2.1
Variables: \(S\) a semigroup, \(s,t\in S\), not necessarily idempotent.
Index: 20
Kind: Col
Prefix Greater: \(s\) and \(t\) are in the same
Suffix Greater: -class

Smaller0: \(t\in V(s)\)
Greater0: \(\mathcal L\)
Page: 107
Section: 2.1
Variables: \(S\) a semigroup
\(s,t\in S\)
Index: 20
Kind: Cor
Prefix Greater: \(st\)
Suffix Greater: \(t\)

Smaller0: \(t\in V(s)\)
Greater0: \(\mathcal L\)
Page: 107
Section: 2.1
Variables: \(S\) a semigroup
\(s,t\in S\)
Index: 20
Kind: Cor
Prefix Greater: \(ts\)
Suffix Greater: \(s\)

Smaller0: \(t\in V(s)\)
Greater0: \(\mathcal R\)
Page: 107
Section: 2.1
Variables: \(S\) a semigroup
\(s,t\in S\)
Index: 20
Kind: Cor
Prefix Greater: \(ts\)
Suffix Greater: \(t\)

Smaller0: \(R(H)\cap L(s)\) and \(R(s)\cap L(H)\) contain an idempotent
Greater0: \(H\) contains an inverse of \(s\)
Page: 107
Section: 2.1
Variables: \(S\) a semigroup,
\(s\in S\)
\(H\) an \(\mathcal H\)-class.
Index: 21
Kind: Prop
Hide middle: yes

Name: Conjugate elements of \(S\)
Definition: Two elements \(uv\) and \(vu\) for some \(u,v\in S^1\).
Page: 107
Chapter: V
Section: 2.1
Variables: \(S\) a semigroup, \(s,t\in S\).

Smaller0: \(s\) and \(t\) are conjugate
Greater0: there exists two elements \(u,v\in S\) such that \(u\mathcal D v\mathcal Ds\mathcal Dt\), \(uv=s\) and \(vu=t\)
Page: 107
Section: 2.1
Variables: \(S\) a semigroup,
\(s,t\in S\).
Index: 22
Kind: Prop
Smaller to greater: iff
Assuming: \(s,t\) are idempotent
Smallest: \(s\mathcal Dt\)

Smaller0: \(xy\) is idempotent
Greater0: \(xyx\) and \(yxy\)
Page: 108
Section: 2.1
Variables: \(S\) a semigroup,
\(x,y\in S\).
Index: 23
Kind: Prop
Hide middle: yes
Greater1: mutually inverse elements
Connect Greater0: are

Name: Regular element of \(S\)
Definition: An element with at least one inverse.
Page: 108
Chapter: V
Section: 2.2
Variables: \(S\) a semigroup

Name: Regular semigroup
Definition: A semigroup whose elements are regular.
Page: 108
Chapter: V
Section: 2.2
Variables: \(S\) a semigroup.

Name: Regular (null)\(\mathcal{D,L,R}\text{ or }\mathcal{J}\)-class
Definition: A \(\mathcal{D,L,R}\text{ or }\mathcal{J}\)-class such that all of its elements have an inverse in \(S\)(otherwise).
Page: 108
Chapter: V
Section: 2.2
Variables: \(S\) a semigroup.

Name: The set of regular elements of \(S\)
Page: 108
Chapter: V
Section: 2.2
Variables: \(S\) a semigroup.
Abbreviation: \(\mathtt{Reg}(S)\)

Smaller0: \(\text{E}(S)\)
Greater0: \(\text{Reg}(S)\)
Page: 108
Section: 2.2
Variables: \(S\) a semigroup.
Smaller to greater: \(\subseteq\)
Greater to smaller: \(\supseteq\)

Name: A regular element
Definition: \(s\overline ss=s\) for some \(\overline s\in S\),
Page: 108
Chapter: V
Section: 2.2
Variables: \(S\) a semigroup
Index: 24
Definition2: \(D(s)\) contains an idempotent
Conjdef: An element \(s\in S\) such that
Definition3: \(L(s)\) or \(R(s)\) contains an idempotent

Name: A regular \(D\)-class
Definition: \(D\) contains a regular element
Page: 109
Chapter: V
Section: 2.2
Variables: \(S\) a semigroup
Index: 25
Definition2: \(D\) contains an idempotent
Conjdef: \(D\) a \(\mathcal D\)-class such that, equivalently
Definition3: there exists \(x,y\in D\) such that \(xy\in D\),
Definition4: each \(\mathcal L\)(each \(\mathcal R\)) class of \(D\) contains an idempotent,

Name: A \(\mathcal K\)-trivial semigroup
Definition: \(x\mathcal Ky\) implies \(x=y\)
Page: 109
Chapter: V
Section: 2.2
Variables: \(\mathcal K\) be \(\mathcal{R,L,H,J}\).
Index: 26
Kind: Prop
Definition2: its regular \(\mathcal K\)-classes are trivial
Conjdef: A semigroup such that, equivalently

Name: An aperiodic semigroup
Definition: A \(\mathcal H\)-trivial finite semigroup,
Page: 109,148?, 202?
Chapter: V, X
Section: 2.2, 2
Index: ,4.22
Kind: ,Prop
Definition2: A finite semigroup whose groups in \(S\) are trivial
Conjdef: A semigroup, equivalently
Definition3: A finite semigroup such that \(x^\omega=x^{\omega+1}\)
Definition4: For each \(x\in S\) there exists an integer \(n\) such that \(x^n=x^{n+1}\)

Smaller0: \(us\mathcal R_S e\mathcal L_S sv\) for some \(e\) idempotent
Greater0: \(s\mathcal L_Tus\mathcal L_Tsvus\)
Page: 109
Section: 2.2
Variables: \(S\) a semigroup
\(T\) a subsemigroup of \(S\)
\(s\in T\),
\(u,v\in T^1\),
Index: 27
Kind: Prop
Smaller1: \(S\) is finite

Smaller0: \(e\in T\)
Greater0: in \(T\) (i.e. \(e\mathcal R_T us\mathcal L_T s\mathcal R_T sv\mathcal L_Te\))
Page: 109
Section: 2.2
Variables: \(S\) a semigroup
\(T\) a subsemigroup of \(S\)
\(s\in T\)
\(e\in S\)
\(u,v\in T^1\),
Index: 27
Kind: Prop
Assuming: \(e\) idempotent
Smaller1: in \(S\) (i.e. \(e\mathcal R_S us\mathcal L_S s\mathcal R_S sv\mathcal L_Se\))

Smaller0: \(sv\mathcal L_Se\mathcal R_S us\), for some idempotent e in T
Greater0: \(s\mathcal R_Tsv\mathcal R_Tsvus\)
Page: 109
Section: 2.2
Variables: \(S\) a semigroup
\(T\) a subsemigroup of \(S\)
\(s\in T\)
\(u,v\in T^1\)
Index: 27
Kind: Prop
Assuming: \(e\) idempotent
Smaller1: \(S\) is finite
Name2: for some idempotent e in T

Page: 110
Chapter: V
Section: 3
Name: The Rees matrix semigroup with \(G\) as structure group, \(P\) as sandwich matrix and \(I,J\) as indexing sets
Set: \(I\times G\times J\),
Variables: \(I,J\) finite sets of index,
\(G\) a group,
\(P\in G^{J\times I}\).
Notation: \(M(G,I,J,P)\)
*: \(((i,g,j),(i',g',j'))\mapsto (i,gp_{j,i'}g',j')\)
0: None
1: In general none

Name: A Brandt semigroup
Definition: with \(I=J\) and
Page: 110
Chapter: V
Section: 3
Index: 1
Definition2: Identity matrix as sandwich matrix
Conjdef: A Rees matrix semigroup

Name: A Brandt aperiodic semigroup
Definition: where the structure group is trivial.
Page: 110
Chapter: V
Section: 3
Index: 1
Conjdef: A Brandt semigroup

Name: The Rees matrix semigroup with 0
Definition: Set \((I\times G\times J)\cup\{0\}\).
Page: 111
Chapter: V
Section: 3
Variables: \(I,J\) finite sets of index,
\(G\) a group,
\(P\in (G\cup\{0\})^{J\times I}\).
Notation: \(M^0(G,I,J,P)\)
Definition2: Product: \(((i,g,j),(i',g',j'))\mapsto\left\{\begin{array}{ll}(i,gp_{j,i'}g',j')&\text{if }p_{j,i'}\ne0\\0&\text{otherwise}\end{array}\right.\)
Conjdef: The semigroup

Name: A binary relation stable on the left
Definition: \(sRt\) implies that \(usRut\) for each \(s,t,u\in M\)
Page: 100
Chapter: V
Section: 1
Variables: \(M\) a monoid
\(R\) a binary relation on \(M\)
Conjdef: A binary relation \(R\) such that

Question: Green relation of the set of \(\left(\begin{array}{ll}a&0\\b&1\end{array}\right)\)
Answer: Single \(\mathcal J\)-class, the other relations are equality.
Page: 103
Chapter: V
Section: 1
Index: 2
Kind: Ex

Page: 98, 98, 100
Definition: \(s=ut\) for some \(u\in S^1\).
Chapter: V
Section: 1
Variables: \(S\) a semigroup
\(s,t\in S\)
Notation: \(s\le_{\mathcal L}t\)
Kind: Prop
Index: 1, 10
Transitive: Stable on the right
Definition2: \(S^1s\subseteq S^1t\)
Definition3: a path from \(t\) to \(s\) in the left Cayley graph of \(M\)

Page: 101, 102
Definition: \(\mathcal L\circ\mathcal R\) or equivalently \(\mathcal R\circ\mathcal L\)
Chapter: V
Section: 1
Notation: \(\mathcal D\)
Kind: Prop, Theo
Index: 8, 9
Definition2: \(\mathcal J\) (assuming finite monoid)
Stable on left: Assume \(x\mathcal Dy\), in this case take \(z\) such that \(x\mathcal Rz\mathcal Ly\), it follows that \(x\mathcal Jz\mathcal Jy\) hence \(x\mathcal Jy\)
Assume \(x\mathcal J y\)
let \(t,u,s,v\) such that \(y=txu\) \(x=syv\), when \(x=(st)^nx(uv)^n\) for each \(n\ge 0\). For some \(n\), \((st)^n\) and \((uv)^n\) are idempotent, thus
\((st)^nx=(st)^n(st)^nx(uv^n)=(st)^nx(uv)^n=x\) and similarly \(x=x(uv)^n\). Thus \(ts\mathcal L\mathcal Rxu\), hence \(y=txy\mathcal Lxu\) and finally \(y\mathcal Dx\)

Page: 98, 99, 103
Definition: \(s\le_{\mathcal L}t\) and \(t\le_{\mathcal L}s\)
Chapter: V
Section: 1
Variables: \(S\) a semigroup and \(s,t\in S\).
Notation: \(s\mathcal Lt\)
Kind: Prop
Index: 1, 10
Transitive: Stable on the right
Definition2: \(s\) and \(t\) belong to the same strongly connected component in the left Cayley Graph.
Definition3: \(S^1s=S^1t\)

Page: 99, 103
Definition: \(s\le_{\mathcal R} t\) and \(t\le_{\mathcal R} s\)
Chapter: V
Section: 1
Variables: \(S\) a semigroup
\(s,t\in S\).
Notation: \(s\mathcal R t\)
Kind: Prop
Index: 10
Transitive: stable on the left
Definition2: \(sS^1=tS^1\)
Definition3: \(s\) and \(t\) belong to the same strongly connected component in the right Caley graph

Smaller0: \(s\mathcal Js^2\)
Greater0: \(H(s)\)
Page: 106
Section: 1
Variables: \(S\) be semigroup
\(s\in S\)
Index: 16
Kind: Cor
Hide middle: yes
Assuming: \(S\) is finite
Greater1: a group
Connect Greater0: is

Name: The Green relations
Definition: \(\mathcal{\le_L,\le_R,\le_H,\le_J}\)
Page: 98
Chapter: V
Variables: \(M\) a semigroup.
Intuition: noncommutative generalisation to semigroups of the standard notion of being a multiple among integers or polynomials.

Page: 98
Definition: \(s=utv\) for some \(u,v\in S^1\)
Chapter: V
Section: 1
Variables: \(S\) a semigroup and
\(s,t\in S\).
Notation: \(s\le_{\mathcal J}t\)
Definition2: \(S^1sS^1 \subseteq S^1tS^1\)

Page: 98
Definition: \(s\le_{\mathcal R}t\) and \(s\le_{\mathcal L}t\)
Chapter: V
Section: 1
Variables: \(S\) a semigroup and
\(s,t\in S\).
Notation: \(s\le_{\mathcal H}t\)

Page: 99
Definition: \(s\mathcal R t\) and \(s\mathcal L t\)
Chapter: V
Section: 1
Variables: \(S\) a semigroup an
\(s,t\in S\).
Notation: \(s\mathcal Ht\)
Definition2: \(s\le_{\mathcal H}t\) and \(t\le_{\mathcal H}s\)

Smaller0: \(S^1sS^1=S^1tS^1\)
Greater0: \(s\mathcal J t\)
Page: 99
Section: 1
Variables: \(S\) a semigroup
\(s,t\in S\).
Smaller to greater: is equivalent to
Hide middle: yes

Page: 98
Definition: (\(s\le_{\mathcal K}t\) and not \(s\mathcal Kt\)).
Chapter: V
Section: 1
Variables: \(S\) a semigroup and
\(s,t\in S\)
\(\mathcal K\) a Green relation
Notation: \(s<_{\mathcal K}t\)

Page: 98, 99, 103
Definition: a path from \(t\) to \(s\) in the right Cayley graph of \(M\)
Chapter: V
Section: 1
Variables: \(A\) a set,
\(M\) an \(A\)-generated monoid
\(s,t\in M\).
Notation: \(s\le_{\mathcal R}t\)
Kind: Prop, Prop
Index: 1, 10
Transitive: stable on the left.
Definition2: \(sS^1 \subseteq tS^1\).
Definition3: \(s=tu\) for some \(u\in S^1\)

Smaller0: \(s\le_{\mathcal R}t\)
Greater0: \(ts=s\)
Page: 101
Section: 1
Variables: \(S\) a semigroup and
\(t\in S\).
Index: 3(1)
Kind: Prop
Smaller to greater: iff
Assuming: \(t\) is an idempotent
Construction: \(s=tu\) for some \(u\in S^1\), thus \(ts=t(tu)=(tt)u=tu=s\)

Smaller0: \(st=s\)
Greater0: \(s\le_{\mathcal L}t\)
Page: 101
Section: 1
Variables: \(S\) a semigroup
\(t,s\in S\).
Index: 1.3(1)
Kind: Prop
Smaller to greater: iff
Hide middle: yes
Assuming: \(t\) is an idempotent
Construction: \(s=ut\) for some \(u\in S^1\), thus \(ss=(ut)t=u(tt)=ut=s\)

Smaller0: \(s\le_{\mathcal R}sxy\)
Greater0: \(s{\mathcal R}sx{\mathcal R}sxy\)
Page: 101
Section: 1
Variables: \(S\) be a semigroup and \(s,x,y\in S\).
Index: 3 (2)
Kind: Prop
Smaller to greater: iff
Construction: \(s\le_{\mathcal R}sxy\), implies \(s\le_{\mathcal R}sxy\le_{\mathcal R}sx\le_{\mathcal R}s\)

Smaller0: \(s\le_{\mathcal H}t\)
Greater0: \(ts=s=st\)
Page: 101
Section: 1
Variables: \(S\) be a semigroup
\(s,t\in S\).
Index: 4
Kind: Prop
Smaller to greater: iff
Hide middle: yes
Assuming: \(t\) is an idempotent
Construction: directly from Prop 3(1) using the fact that \(\le_{\mathcal H}=\le_{\mathcal R}\cap\le_{\mathcal L}\)

Smaller0: \(S\) is finite
Greater0: \(us\mathcal H s\)
Page: 103
Section: 1
Variables: \(S\) a semigroup,
\(s,u,v\in S\).
Index: 9(5)
Kind: Theo
Hide middle: yes
Smaller1: \(s=usv\)
Greater1: \(s\mathcal H sv\)
Construction: \(s=usv\), thus \(s\le_{\mathcal J}us\), by prop (1), \(s\mathcal R sv\), and since its stable on the left \(us\mathcal R usv=s\).
By symmetry \(s\mathcal Lus\), \(sv\mathcal L usv=s\)
thus \(us\mathcal Hs\mathcal H sv\)

Smaller0: \(t\) is a weak inverse of \(s\)
Greater0: \(ts\)
Page: 106
Section: 2.1
Variables: \(S\) a semigroup
\(s,t\in S\).
Index: 19
Greater1: idempotent
Connect Greater0: is
Counterexample: \(\langle\{t,s\},\{tsts=ts,stst=st\}\rangle\)

Smaller0: \(S\) is finite
Greater0: \(\mathcal J\)-class of the identity
Page: 106
Section: 1
Variables: \(S\) a monoid.
Index: 17
Hide middle: yes
Prefix Greater: the
Greater1: group
Connect Greater0: is

Smaller0: \(D\) contains an idempotent
Greater0: each \(\mathcal R\)- and each \(\mathcal L\)-class of \(D\) contains an idempotent
Page: 104
Section: 1
Variables: \(S\) a semigroup
\(D\) be a \(\mathcal D\)-class of \(S\).
Index: 12
Kind: Prop
Smaller to greater: iff
Construction: \(e,s\in D\), \(e\) idempotent
\(e\mathcal Rr\mathcal Ls\) for some \(r\in S^1\).
\(e\mathcal Rr\) and \(e\) idempotent implies \(er=r\) by prop 3(1), and \(e=ru\) for some \(u\)
\(urur=u(ru)r=uer=ur\), thus \(ur\) is idempontent
\(r(ur)=er=r\) thus \(r\mathcal L ur\)
\(s\mathcal r\mathcal L ur\), thus the \(\mathcal L\) class of \(s\) contains an idempotent

Smaller0: \(A,B\) are included in the same \(\mathcal D\)-class
Greater0: \(A,B\) are isomorph
Page: 105
Section: 1
Variables: \(S\) a semigroup
\(A,B\subseteq S\).
Index: 14
Smaller1: \(A,B\) are maximal subgroups

Smaller0: \(t\) is a weak inverse of \(s\)
Greater0: \(sts\)
Page: 106
Section: 2.1
Variables: \(S\) a semigroup,
\(s,t\in S\).
Index: 19
Greater1: an inverse of \(t\)
Connect Greater0: is
Extra: s aussi est un inverse de t ?

Smaller0: \(t\) is a weak inverse of \(s\)
Greater0: \(\mathcal L(t)\)
Page: 106
Section: 2.1
Variables: \(S\) a semigroup,
\(s,t\in S\).
Index: 19
Hide middle: yes
Prefix Greater: \(\mathcal L(st)=\)

Smaller0: \(t\) is a weak inverse of \(s\)
Greater0: \(\mathcal L(sts)\)
Page: 106
Section: 1
Variables: \(S\) a semigroup
\(s,t\in S\).
Index: 19
Greater1: \(\mathcal L(ts)\)
Connect Greater0: equals

Smaller0: \(x\to xp\)
Greater0: \(x\to xq\)
Page: 81
Section: 1
Variables: \(S\) a semigroup,
\(s,t\in S\),
\(p,q\in S^1\).
Index: 10
Kind: Prop
Smaller to greater: is the inverse of
Assuming: \(s=tp\) and \(t=sq\).

Smaller0: \(S\) is finite
Greater0: \(\le_{\mathcal J}\)-maximal \(\mathcal J\)-class of \(S\)
Page: 106
Section: 1
Variables: \(S\) be a semigroup.
Index: 17
Hide middle: yes
Prefix Greater: the
Greater2: regular
Connect Greater1: are either
Greater3: reduced to a single null element
Connect Greater2: or

Smaller0: \(R\) is regular
Greater0: every row and every column of the sandwich matrix of \(R\) has a nonzero entry
Page: 111
Section: 3
Variables: \(R\) a rees matrix semigroup with zero
Index: 28
Smaller to greater: iff
Hide middle: yes
Context: Rees matrix

Question: The number of inverse of \(s\) in \(H\) is
Answer: at most one.
Page: 108
Chapter: V
Section: 4.1
Variables: \(S\) a semigroup
\(s\in S\)
\(H\) a \(\mathcal H\)-class.
Index: 21
Kind: Prop
Construction: \(t_1,t_2\) be inverse of \(s\)
\(st_1\) and \(st_2\) are idempotent of the same \(\mathcal H\)-class by Cor 20
, thus equal
by symmetry
\(t_1s=t_2s\)
Thus \(t_1st_1=t_2st_1=t_2st_2\) thus \(t_1=t_2\)
what ??
?

Name: A shuffle ideal of \(A\)
Definition: of the form [$]M\textvisiblespace\textvisiblespace A^*[/$] for some language \(M\)
Page: 145
Chapter: VII
Section: 2
Variables: \(A\) alphabet
Index: , 7, 8, 10
Kind: ,Theo, Cor, Theo
Definition2: finite union of simple languages
Conjdef: A language \(L\subseteq A^*\), equivalently
Definition3: of the form [$]M\textvisiblespace\textvisiblespace A^*[/$] for some finite language \(M\)
Definition4: such that [$]L=L\textvisiblespace\textvisiblespace A^*[/$]
Definition5: the syntactic ordered monoid of \(L\) satisfies \(1\le x\)
Property: recognisable
Construction: \(L\) is the set of minimal words of the language

Name: A simple language
Definition: [$]a_1\dots a_n\textvisiblespace\textvisiblespace A^*[/$] for some \(a_1,\dots,a_n\in A\)
Page: 15
Chapter: VII
Section: 2
Variables: \(A\) alphabet
Definition2: \(A^*a_1\dots A^*a_nA^*\) for some \(a_1,\dots,a_n\in A\)
Conjdef: A language of the form, equivalently
Definition3: The set of superwords of \(a_1\dots a_n\) for some \(a_1,\dots,a_n\in A\)