Re: Structure of a semigroup

From: Riivo Must <riivo@xxxxxxxxxxx>
Newsgroups: sci.math
Subject: Structure of a semigroup

> I defined a new semigroup called 2-inverse semigroup: it's the one
> where every element has precisely 2 inverse elements (remind that b
> is inverse to a iff a=aba and b=bab). I've found so far only some
> elementary facts: it doesn't have 0 nor 1, it is orthodox, it doesn't
> have any idempotent which commutes with every other idempotent

> If somebody can give some hints how to find out what this 2-inverse
> semigroup actually is and what is it's structure (eg. what do
> idempotents form?), I'd be thankful.

Huh? There isn't just one 2-inverse semigroup.

Consider free semigroup with two generators, a,b and relators
a = aba; b = bab; ab = ba

This semigroup has only the elements
ab, a^n, b^m, n,m in N

This is consistent as either of the first two relators with the third
reduce the word
a^n b^m to a^(n-1) b^(m-1)

ab = ab.ab is the only idempotent and commutes with everybody.

What's orthodoxy?

> (remind that in inverse semigroup, where every element has precisely
> 1 inverse element, all idempotents commute with each other).

They do?

Note: a=bc+d<=r is hard to read.
a = bc + d < r for being easier to
read, is more likely to get a reply.

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