Questions tagged [inverse-semigroups]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3
votes
1answer
72 views

$C^*$-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
0answers
81 views

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 $\mathcal J$ ...
2
votes
0answers
70 views

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 ...
-2
votes
3answers
41 views

Properties of groups with integer exponents

For any elements a and b from a group $\lt A,\ * \gt$ and any integer n, prove that $\begin{align} & (a^{-1}\cdot b\cdot a)^n = a^{-1}\cdot b^n\cdot a \end{align}$ This is what I tried: LHS: $\...
0
votes
0answers
44 views

Free regular semigroups!

Can someone help me to understand how the free regular semigroups are defined? I've been looking this for hours, but I just can't get it! Any help is greatly appreciated.
2
votes
1answer
56 views

Showing $|\mathcal{I}_X|=\sum_{r=0}^n{n \choose r}^2r!$.

This is Exercise 5.11.3 of Howie's "Fundamentals of Semigroup Theory". Definition 1: Let $X$ be a set. The symmetric inverse semigroup, denoted $\mathcal{I}_X$, consists of all partial one-to-one ...
6
votes
4answers
739 views

Definition of groups

This seems like a very basic question but got me confused. When defining a group we introduce the unit element $e$ which has the following property $$ge = eg = g \quad \forall g\in G$$ and then the ...
2
votes
1answer
85 views

Is this semigroup a group?

I have the following structure $$\langle a,b\mid a^5=a, b^9=b, a^2=b^2, ab=b^7a\rangle$$ GAP tells me that this is of order $8$. Is it probably the quaternion group? How could GAP show it? Thanks!
-2
votes
1answer
78 views

Help with understanding the “upper saturation/closure” of a subset of an inverse Semigroup.

I have been working through Howie's book on semigroups, and while doing so I've come across this question: "For every subset $K$ of an inverse semigroup $S$, and for every $s \in S$, show that $(Ks)ω ...
0
votes
1answer
88 views

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 ...
2
votes
1answer
114 views

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
0answers
102 views

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 ...
3
votes
2answers
68 views

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
2answers
60 views

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 ...
0
votes
0answers
49 views

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{...
5
votes
1answer
129 views

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: ...
0
votes
1answer
180 views

S an inverse semigroup with semilattice of idempotents E and $\sigma$ the minimum group congruence on S.

Let S be an inverse semigroup with semilattice of idempotents E, and let $\sigma$ be the minimum group congruence on S. Show that the following statements are equivalent: (a) $x\sigma y$; (b) $(\...
4
votes
1answer
177 views

D-􀀀classes in an inverse semigroup are ‘square’.

Show that the $\mathcal{D}$-classes in an inverse semigroup are ‘square’. More precisely, show that there is a bijection from the set of $\mathcal{L}$-classes in a $\mathcal{D}$-class $D$ onto the set ...
1
vote
0answers
91 views

Congruence on inverse semigroup

Could you please help me to understand the reason why we are interested in trace of a congruence and the kernel's congruence when we're talking about the congruences on inverse semigroup. also I have ...
0
votes
1answer
37 views

Left and Right Inverses with semigroups

Having the semigroup $(F,\circ)$ where $F=\{f: f: \mathbb{N}\to \mathbb{N}, \mathrm{Dom}(f) = \mathbb{N}\}$. The identity $e∈F$ is the function $e(n) = n$, define the function $g(n) = m$ if $5(m-1)&...
2
votes
1answer
510 views

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 ...
5
votes
1answer
279 views

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 ...
2
votes
4answers
675 views

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 ...
2
votes
2answers
148 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 ...
3
votes
1answer
142 views

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 ...
1
vote
1answer
138 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
101 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, please ...
4
votes
1answer
563 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
46 views

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....
4
votes
0answers
119 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
918 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 $E_S$ ...