# Questions tagged [inverse-semigroups]

For questions about inverse semigroups, Clifford semigroups and other classes of semigroups with the notion of inverses from semigroup theory. Use in conjunction with the tag (semigroups).

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### Schutzenberger graphs of an Inverse Semigroup?

I recently came across the idea of extending the well-known Cayley graph construction for semigroups and learned that the outcome does not have all the expected properties even for the nice classes of ...
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### Why does this chart not include invertibility for inverse semigroups?

This chart shows that inverse semigroups, have associativity and divisibility, but not invertibility. At first this made me think that they might not have invertibility despite the word "inverse&...
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### $C^*$-algebra norm computations

I'm fairly new to $C^*$-algebras and Hilbert space. Given the algebraic relations of the $C^*$-algebra, I am having a lot of trouble computing the norm of its elements and am wondering if there are ...
1 vote
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### Principal factors of an inverse semigroup $S$ are Brandt semigroup except for the kernel, which is a group

This is the exercise problem of John.M. Howie book 5.4 Let $S$ be an semigroup and $J(a)$ is denoted as a principal ideal generated by $a$, where $a \in S.$ Two element $a, b$ are $\mathcal J$ ...
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### Munn semigroup and their properties

Some definition : Let $S$ be an inverse semigroup and $E$ be the set of all idempotents in $S$. Then $E$ form a semilattice under this relation $$e \leq f \; \; \text {iff} \; ef = fe = e$$ Define a ...
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### In the symmetric inverse semigroup, $I_X$, if $|X| = n$, show that $|I_X| = \sum_{r=0}^n \begin{pmatrix} n\\ r\end{pmatrix}^2 r!$

In the symmetric inverse semigroup, $I_X$, if $|X| = n$, show that $$|I_X| = \sum_{r=0}^n \binom{n}{r}^2 r!$$ This is a question form Howie's book on semigroup theory, unfortunately there isn't a ...
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### Example of a locally inverse semigroup which isn't a generalized inverse semigroup

I'm studying Howie's Fundamentals of Semigroup Theory. A semigroup $S$ is locally inverse if $eSe$ is inverse for any idempotent $e$ of $S$. A semigroup is a generalized inverse semigroup if is ...
1 vote
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### Prove the following characterization for inverse semigroups

I'm trying to prove the following characterization for inverse semigroups: Let $S$ be a regular semigroup. For each $a \in S$ define $A(a) = \{x \in S : a = axa \}.$ The semigroup $S$ is inverse if ...
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### Ways of describing/classifying a strange structure where $a\ast a = b; b\ast b = a; a\ast b = a$

Problem is mostly in the title. I'm a complete amateur when it comes to math. I was doodling on graph paper when I found structure that I'm having a hard time wrapping my head around. The operation ...
1 vote
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### How is $ss^{-1}$ idempotent in an inverse monoid?

An inverse monoid S is a monoid such that for all $s \in S$, there exists a $t \in S$ such that $s=sts$ and $t=tst$. In this case, we write $t = s^{-1}$. Why is $ss^{-1}$ an idempotent? I don't ...
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### Generators and relations of symmetric inverse semigroup $I_{3}$

Let $I_{3}$ be an inverse semigroup consisting of all partial bijections on a set $\{1,2,3\}$, called the symmetric inverse semigroup. Then \begin{align*} I_{3}=\left\{\emptyset, \binom{1}{1}，\binom{...
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### GAP tells this semigroup is not a group.

Happy Nowruz 2016 to every one here! Using the code which James pointed here; I was playing with the following finite semigroup: ...
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### Do the idempotents in an inverse semigroup commute?

I have been looking at this for hours now. Why is it true that idempotents of an inverse semigroup commute? It seems like this should be straightforward but I just can't get it. Any help is greatly ...
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### A reflective subcategory of the category of inverse semigroups.

The Question. I'm reading Lawson's "Inverse Semigroups: The Theory of Partial Symmetries" and I've hit something I don't understand. It's claimed on page 34 of my copy that The category of ...
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### Example of an inverse semigroup

An inverse semigroup $S$ is a semigroup in which for each $x\in S$ there exists a unique $y\in S$ such that $xyx=x$ and $yxy=y$. I'm trying to find an explicit example(which is not a group) of such ...
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### 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 ...
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### Is there a name for the generalization of the concept "Abelian group" where the axiom $-x+x = 0$ is weakened to the following?

Is there a name for the generalization of the concept "Abelian group" where the axiom $−x+x=0$ is replaced by the following list? $−0=0$ $−(x+y)=−x+−y$ $−(−x)=x$ $x+(-x)+x = x$ In multiplicative ...
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### 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 ...
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### 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, please ...
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### 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 $G \cup \{0\}$ ...
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### quasiperiodic tilings: inverse semigroup -- non-commutative geometry connect?

A Connes' Noncommutative Geometry (1994) and M Lawson's Inverse Semigroups (1998) contain sections on quasiperiodic tilings, yet as far as I can tell neither seems to refer to the other field of study....
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### 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 ...
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### 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 $E_S$ ...