Title: Theory / Unimportant / CS / Fundamental / Automata / Finite / Pin{ / D Advanced tools= / 16 Relational morphisms+32
Tags: closed fromMath hasSeenConjDef récupéré

Name: Subsemigroups of \(S\)
Page: 24, 228?
Section: 3.4, 1
Chapter: II, XIII
Index: 3, 4
Variables: \(S\) a semigroup
Kind: , Prop
Under: image by morphism
Condition: relational
Under2: inverse image by morphism

Name: A relational morphism of semigroup
Definition: A relation \(\tau:S\to T\) which satisfies:
-\(\tau(s)\) is non empty for each \(s\in S\)
-\(\tau(s_1)\tau(s_2)\subseteq\tau(s_1s_2)\)
Page: 227, 228
Chapter: XIII
Section: 1
Variables: \(S,T\) semigroups
Index: , 3
Kind: , Prop
Definition2: \(\beta\circ\alpha^{-1}\), for \(\alpha:R\to S\), \(\beta:R\to T\) morphisms, with \(\alpha\) surjective and \(R\subseteq S\times T\)
Example: morphisms
Example2: inverse of surjective morphism

Name: A relational morphism of monoid
Definition: \(\tau\) is a relational morphism as semigroup
Page: 227
Chapter: XIII
Section: 1
Variables: \(S,T\) monoid
Kind: Cor
Definition2: \(1_T\in\tau(1_S)\)
Conjdef: A relation \(\tau:S\to T\) which satisfies

Name: The graph of \(\tau\)
Definition: the set of \((s,t)\) such that \(t\in\tau(s)\) and \(s\in S\)
Page: 227
Chapter: XIII
Section: 1
Variables: \(S,T\) semigroup
\(\tau:S\to T\) a relational morphism
Kind: Prop
Typ: A subsemigroup/submonoid of \(S\times T\)

Name: Relational morphism
Page: 227
Section: 1
Chapter: XIII
Index: 2
Kind: Prop
Under: composition

Name: The canonical factorisation of \(\tau\)
Definition: [$]\alpha:\defFun{R}{S}{(s,t)}{s}[/$]
Page: 228
Chapter: XIII
Section: 1
Variables: \(S,T\) semigroups
\(\tau:S\to T\) relational morphisms
Index: 3
Kind: Prop
Definition2: [$]\beta:\defFun{R}{T}{(s,t)}{t}[/$]
Conjdef: the pair
Applied to3: Two morphisms, such that \(\tau=\beta\circ\alpha^{-1}\) with \(\alpha\) surjective