Title: Theory / Unimportant / CS / Fundamental / Automata / Finite / Pin{ / C The profinite world / 12 Equations and languages=
Tags: hasSeenConjDef

Name: A profinite equation
Definition: A pair of profinite words
Page: 211
Chapter: VIII
Section: 1
Variables: \(A\) an alphabet
Denoted by: \(u\to v\)

Name: An explicit equation
Definition: A profinite equation whose words are finite
Page: 211
Chapter: VIII
Section: 1
Variables: \(A\) an alphabet

Name: \(L\) satisfies \(u\to v\)
Definition: \(u\in\bar L\) implies \(v\in\bar L\)
Page: 211
Chapter: VIII
Section: 1
Variables: \(A\) an alphabet
\(u\to v\) a profinite equation
\(L\subseteq A^*\)
Kind: Cor
Definition2: \(\hat\eta(u)\in\eta(L)\) implies \(\hat\eta(v)\in\eta(L)\) for \(\eta\) the syntactic morphism
Definition3: \(\hat\phi(u)\in\phi(L)\) implies \(\hat\phi(v)\in\phi(L)\) for \(\phi\) any morphism onto a finite monoid recognising \(L\)

Name: The set defined by \(E\)
Definition: containing languages satisfying the equations of \(E\)
Page: 212
Chapter: VIII
Section: 2
Variables: \(A\) an alphabet
\(E=\{u_i\to v_i\}\) a set of profinite equation
Index: , 6
Kind: , Prop
DefType: The lattice of languages

Name: A lattice of languages of \(A^*\)
Definition: containing the empty language, the full language, closed under finite union and finite intersection
Page: 212, 213
Chapter: VIII
Section: 2
Variables: \(A\) an alphabet
Index: , 5, 6
Kind: , Cor, Theo
Definition2: defined by a set of explicit equation (assuming it is a finite set of languages)
Conjdef: A set of languages of \(A^*\), equivalently
Definition3: defined by a set of explicit equation (assuming its languages are recognisable)

Definition: The set of pair of profinite words \((u,v)\in\hat{A^*}\times\hat{A^*}\) satisfying \(u\to v\)
Page: 212
Chapter: VIII
Section: 2
Variables: \(A\) an alphabet
\(L\) a language
Notation: \(E_L\)
Index: 6
Kind: Theo

Definition: \(u\to v\) and \(v\to u\)
Page: 214
Chapter: VIII
Section: 2
Variables: \(A\) an alphabet
\(L\) a language
\(u,v\) profinite words
Notation: \(u\leftrightarrow v\)
Index: 6

Name: A Boolean algebra of languages
Definition: A set of languages which can be defined by a set of equations of the form \(u\leftrightarrow v\) with \(u,v\) profinite words
Page: 214
Chapter: VIII
Section: 2
Variables: \(A\) an alphabet
\(L\) a language
Index: 8
Kind: Cor