−Table of Contents
Inverse semigroups
Abbreviation: InvSgrp
Definition
An \emph{inverse semigroup} is a structure S=⟨S,⋅,−1⟩ such that
⋅ is associative: (xy)z=x(yz)
−1 is an inverse: xx−1x=x and (x−1)−1=x
idempotents commute: xx−1yy−1=yy−1xx−1
Morphisms
Let S and T be inverse semigroups. A morphism from S to T is a function h:S→T that is a homomorphism:
h(xy)=h(x)h(y), h(x−1)=h(x)−1
Examples
Example 1: ⟨IX,∘,−1⟩, the \emph{symmetric inverse semigroup} of all one-to-one partial functions on a set X, with composition and function inverse. Every inverse semigroup can be embedded in a symmetric inverse semigroup.
Basic results
x∗x=x⟹∃y x=y∗y−1
∀x∃y xx−1=y−1y
Properties
Finite members
f(1)=1f(2)=2f(3)=5f(4)=16f(5)=52f(6)=208f(7)=911f(8)=4637f(9)=26422f(10)=169163f(11)=1198651f(12)=9324047f(13)=78860687f(14)=719606005f(15)=7035514642