1
vote
1answer
35 views

Involution on inverse semigroups

I'm trying to prove the following for inverse semigroups $\bf Def:$ an inverse semigroup $S$is a semigruop such that for each $x\in S$ the exists a unique $y\in S$ such that $xyx=x$ and $yxy=y$. An ...
1
vote
1answer
60 views

Is the set of all uniquely invertible elements of a semigroup an inverse subsemigroup?

For a semigroup $S$ and $x\in S$, an element $y\in S$ is called an inverse of $x$ iff $xyx=x$ and $yxy=y$. $S$ is called an inverse semigroup when every element of $S$ has a unique inverse. Every ...
0
votes
1answer
60 views

Please check my solution, inverse semigroups.

Let $x$ and $y$ be elements of an inverse semigroup $S$. Then the following are equivalent: (1) $xy^{-1}x = x$ (2) $x^{-1}yx^{-1}=x^{-1}$ I'm not sure in my solution, ...
2
votes
1answer
179 views

Rees matrix semigroup

Let $\mathcal{M}^0 = \mathcal{M}^0 ( G; I , \Lambda ;P)$ be a Rees matrix semigroup ($G$ a group, $I$, $\Lambda$ non-empty sets, $P=(p_{\lambda i})$ a $\Lambda \times I$ matrix over $G \cup \{0\}$ ...
3
votes
0answers
73 views

Is the universal inverse semigroup of a commutative semigroup an embedding?

The question of existence of a universal inverse semigroup of an arbitrary semigroup has been answered before (this is a construction similar to the Grothendieck group). Let's refer to the universal ...
3
votes
1answer
302 views

Description of Green's relations $\mathcal{L}$, $\mathcal{R}$ in an inverse semigroup

Theorem: Let $S$ be an inverse semigroup, and let $x,y\in S$ and $e,f\in E_{S}$ then $x\mathcal{L}y$ if and only if $x^{-1}x=y^{-1}y$ $x\mathcal{R}y$ if and only if $xx^{-1}=yy^{-1},$ where $E_S$ ...