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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: xx1x=x and (x1)1=x

idempotents commute: xx1yy1=yy1xx1

Morphisms

Let S and T be inverse semigroups. A morphism from S to T is a function h:ST that is a homomorphism:

h(xy)=h(x)h(y), h(x1)=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

xx=xy x=yy1

xy xx1=y1y

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

http://oeis.org/A001428

Subclasses

Superclasses

References


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