Title: Theory / Unimportant / CS / Fundamental / Automata / Finite / Pin{ / D Advanced tools= / 18 Wreath product+32
Tags: hasSeenConjDef defToAlgebra defToAlgebra2 setMultiplication

Name: A left semigroup action of \(T\) on \(S\)
Definition: \(t_1\cdot (t_2\cdot s)=(t_1t_2)\cdot s\)
Page: 243
Chapter: XV
Section: 1
Variables: \((T,.)\) a semigroup
\(S\) a set
Definition2: \(t\cdot(s_1+s_2)=t\cdot s_1+t\cdot s_2\)
Conjdef: A map \(\cdot:T^1\times S\to S\) such that, for all \(s_i\in S\), \(t_i\in T\)

Name: A left monoid action of \(T\) on \(S\)
Definition: \(\cdot\) is a left semigroup action
Page: 243
Chapter: XV
Section: 1
Variables: \((T,.,1)\) a monoid
\(S\) a set
Definition2: \(1\cdot s=s\)
Conjdef: A map \(\cdot:T^1\times S\to S\) such that, for all \(s_i\in S\), \(t_i\in T\)

Name: A unitary left semigroup action of \(T\) on \(S\)
Definition: A semigroup action of \(T\) on \(S\) such that \(t\cdot 0=0\) for each \(t\in T\)
Page: 243
Chapter: XV
Section: 1
Variables: \((T,.)\) a semigroup
\((S,+,0)\) a monoid

Page: 243
Chapter: XV
Section: 1
Name: The semidirect product of \(S\) and \(T\) with respect to \(\cdot\)
Set: \(S\times T\)
Variables: \((T,.)\) a semigroup
\((S,+)\) a semigroup
\(\cdot\) a left action of \(T\) on \(S\)
Notation: \(S\star T\)
*: \((s,t)(s',t')\mapsto (s+t\cdot s',tt')\)
1: \((1_S,1_T)\) if they exists
Context: Semigroup

Page: 243
Chapter: XV
Section: 2
Name: The wreath product of \(X\) and \(Y\)
Set: \(P\times Q\)
Variables: \(X=(P,(S,+_S),+_{P,S})\)
\(Y=(Q,(T,._P), ._{Q,T})\) two transformation semigroups with right actions
Notation: \(X\circ Y\)
+: as in \(S\circ T\)
Typ: The transformation semigroup
×: right action: \((p,q)\circ(f,t)\mapsto (p+f(q),q\cdot_{Q,T}t)\)
Extra: 0 to \times

Name: The wreath product of \(S\) and \(T\)
Definition: \(S^{T^1}\star T\)
Page: 244 https://en.wikipedia.org/wiki/Wreath_product
Chapter: XV
Section: 2
Variables: \((S,+_S)\)
\((T,._P)\) two semigroups
Notation: \(S\circ T\)
Index: 1
Kind: Prop