Title: Theory / Unimportant / CS / Fundamental / Automata / Infinite / Perrin Pin+10 / 02 semigroup
Tags: fromMath mathToDef hasSeenConjDef toBinaryRelation typToBinary

Name: A factorization of \(u\)
Definition: A sequence \((u_n)_{n\ge0}\) of words of \(A^+\) such that \(u=u_0u_1\dots\).
Page: 77
Chapter: II
Section: 2.1
Variables: \(A\) an alphabet
\(u\in A^\omega\)

Name: A subfactorization of \((u_n)_{n\ge0}\)
Definition: A factorization \((u'_n)_{n\ge0}\) such that there exists a strictly increasing sequence of strictly positive integers \((k_n)_{n\ge 0}\) such that \(u_{i}=u'_{k_i}\dots u_{k_{n-1}-1}\).
Page: 77
Chapter: II
Section: 2.1
Variables: \(A\) an alphabet, \(u\in A^\omega\). Let \((u_n)_{n\ge0}\) a factorization of \(u\).

Name: \(u\) is Ramseyan for \(\phi\)
Definition: There exists an element \(e\in E\) such that, for each \(i\le j\), \(\phi(y_i\dots y_j)=e\).
Page: 77
Chapter: II
Section: 2.1
Variables: \(A\) an alphabet
\(\phi:A^+\to E\)
\(xy_0y_1\dots\) a factorization of \(u\).

Question: Every infinite word of \(A^\omega\) admits
Answer: a Ramseyan factorization for \(\phi\).
Page: 77
Chapter: II
Section: 2.1
Variables: \(A\) an alphabet,
\(E\) a set,
\(\phi:A^+\to E\)
Index: 1
Kind: Theo
Assuming: \(E\) is finite

Smaller0: \(E\) is finite
Greater0: \(xy_0y_1\dots\) such that \(\phi(y_i)=\phi(y_j)\) and \(\phi(x)\phi(y_i)=\phi(y_i)\) for all \(i<j\)
Page: 78
Chapter: II
Section: 2.1
Variables: \(A\) an alphabet,
\(E\) a semigroup
\(\phi:A^+\to E\)
Index: 2
Kind: Theo
Hide middle: yes
Prefix Greater: every infinite word of \(A^\omega\) admits a factorization

Smaller0: \(\phi\) is a morphism of semigroup, \(E\) is finite
Greater0: a subfactorization \(x'y'_0y'_1\dots\) such that \(\phi(y'_i)=\phi(y'_j)\) is idempotent and \(\phi(x)\phi(y'_i)=\phi(y'_i)\) for all \(i<j\)
Page: 78
Chapter: II
Section: 2.1
Variables: \(A\) a finite alphabet, \(E\) a semigroup \(\phi:A^+\to E\), \(u\in A^\omega\) and \(y_1y_2\dots\) a factorization of \(u\)
Index: 3
Kind: prop
Hide middle: yes
Prefix Greater: \(u\) admits

Question: There exists an increasing sequence \((k_i)_{i\in\mathbb N} \) such that
Answer: \(s_{k_n}\dots s_{k_{n+1}-1}=s_{k_{n'} }\dots s_{k_{n'+1}-1}\) idempotent, and \(s_0\dots s_{k_0-1}s_{k_n}\dots s_{k_{n+1}-1}=s_{k_n}\dots s_{k_{n+1}-1}\) for all \(n<n'\).
Page: 78
Chapter: II
Section: 2.1
Variables: \(S\) a semigroup, \((s_n)_{n\in\mathbb N}\) in \(S\).
Index: 4
Kind: Cor
Assuming: \(S\) is finite

Name: An element linked to \(e\)
Definition: An element \(s\in S\) such that \(se=s\).
Page: 79
Chapter: II
Section: 2.2
Variables: \(S\) a semigroup,
\(s\in S\)
\(e\in E(S)\)

Name: A linked pair of \(u\)
Definition: A linked pair \((s,e)\) such that there exists a factorization of \(u\) associated with \((s,e)\)
Page: 79
Chapter: II
Section: 2.2
Variables: \(S\) a semigroup,
\(\phi:A\to S\) a semigroup morphism
\(A\) an alphabet
\(u\in A^\omega\).

Name: A factorization associated with \((s,e)\)
Definition: \(\phi(u_0)=s\)
Page: 79
Chapter: II
Section: 2.2
Variables: \(S\) a semigroup,
\((s,e)\) a linked pair,
\(A\) an alphabet
\(\phi:A\to S\) a semigroup morphism
\(u\in A^\omega\).
Definition2: each \(\phi(u_i)=e\) for \(i>0\)
Conjdef: A factorization \(u_0u_1\dots\) of \(u\) such that

Name: An infinite word associated with \((s,e)\)
Definition: \(u\in A^\omega\) such that there exists a factorization \(u_0u_1\dots\) of \(u\) associated with \((s,e)\).
Page: 79
Chapter: II
Section: 2.2
Variables: \(S\) a semigroup,
\((s,e)\) a linked pair,
\(A\) an alphabet
\(\phi:A\to S\) a semigroup morphism

Name: A \(\phi\)-simple subset
Definition: A set of the form \(\phi^{-1}(s)\left(\phi^{-1}(e)\right)^\omega\) for \((s,e)\) a linked pair.
Page: 79
Chapter: II
Section: 2.2
Variables: \(S\) a semigroup,
\(A\) an alphabet
\(u\in A^\omega\)
\(\phi:A^+\to S\) a semigroup morphism
Name2: A simple subset

Name: The group generated by \(\langle S\mid l_1=r_1,\dots,l_m=r_m\rangle\)
Definition: The quotient of \(S^*\) by the group generated by the \(l_i r_i^{-1}\).
Page: 26
Chapter: 1
Section: 2
Variables: \(\langle S\mid l_1=r_1,\dots,l_m=r_m\rangle\) be a presentation of the group

Smaller0: \((s,e)\) is a linked pair of \(xu\)
Greater0: \(s=rt\)
Page: 79
Chapter: II
Section: 2.2
Variables: \(S\) a semigroup,
\(A\) an alphabet and
\(\phi:A^+\to S\) a semigroup morphism
\(u\in A^\omega\),
\(x\in A^*\)
\((s,e)\in S\times S\)
Index: 5
Kind: Prop
Smaller to greater: iff
Prefix Greater: there is \(t\in S\) such that
Greater1: \((t,e)\) is a linked pair of \(u\)

Page: 79, 80, 81
Definition: \(s_2=s_1x_1\), \(e_1=x_1x_2\) and \(e_2=x_2x_1\) for some \(x_1,y_2\in S^1\)
Chapter: II
Section: 2.2
Variables: \(S\) a semigroup,
\((s_i,e_i)\in S\times E(S)\)
Name: \((s_1,e_1)\) and \((s_2,e_2)\) are conjugate
Kind: , Prop, Prop, Cor
Index: 6, 7, 9
Definition2: \(s_2=s_1x\), \(xe_2\mathcal Re_1\) and \(e_1\mathcal D e_2\) for some \(x\in S^1\)
Equivalence: Equivalence relation
Definition3: \(\phi^{-1}(s_1)(\phi^{-1}(e_1))\) intersects \(\phi^{-1}(s_2)(\phi^{-1}(e_2))\) for each morphism \(\phi:A^+\to S\) with \(S\) finite

Smaller0: the \((s_i,e_i)\) are conjugate
Greater0: \(s_1\mathcal R s_2\)
Page: 79
Chapter: II
Section: 2.2
Variables: \(S\) a semigroup,
\((s_i,e_i)\in S\times E(S)\)
Smaller1: \(e_1\mathcal D e_2\)

Name: Conjugate factorization of \(u\)
Definition: The relation on factorization of \(u\), containing the pair of the form \((v_0,(v_{2i-1}v_{2i})_{i\in\mathbb N_{>0}})\) and \((v_0v_1,(v_{2i}v_{2i+1})_{i\in\mathbb N_{>0}}))\) for \((v_i)\) a factorization of \(u\), such that each \(\phi(v_{2i-1}v_{2i})\) are equal and each \(\phi(v_{2i}v_{2i+1})\) are equal.
Page: 80
Chapter: II
Section: 2.2
Variables: \(A\) an alphabet,
\(u\in A^\omega\)

Smaller0: the \((u_i^j)\) are conjugate factorizations
Greater0: the \((u_i^j)\)'s are associated with conjugate linked pairs
Page: 80, 81
Chapter: II
Section: 2.2
Variables: \(A\) an alphabet,
\(u\in A^\omega\),
\((u_i^j)_{i\in\mathbb N}\) factorizations of \(u\) for \(j\) being 1 or 2
Index: 8
Kind: Prop
Smaller to greater: iff