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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 ...
Bumblebee's user avatar
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Why do we have $s.\chi_x=\chi_x$ implies that $s(x)=x u$ for some $u \in \mathfrak{C}^{*, 0}$ in an inverse semigroup?

This is a detail in Theorem 4.4 on Page 9 of Xin Li's paper Left Regular Representations of Garside Categories I. C-star-Algebras and Groupoids. In the proof he said that: $$s.\chi_x=\chi_x \text{ ...
ScienceAge's user avatar
1 vote
1 answer
47 views

Is multiplication by a fixed element an open map in a topological semigroup?

In a topological semigroup $G$, multiplying by a fixed element is continuous as we can decompose it into $$G \overset{\text{diagonal}}{\longrightarrow} G\times G \overset{\text{const}\times\text{id}}{\...
Bumblebee's user avatar
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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&...
SMMH's user avatar
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I have question about a semigroup for a Cauchy problem

Let $(A, D(A))$ be the generator of a strongly continuous semigroup $(S(t))$ on $X$. Then the following Cauchy problem $$ \left\{\begin{array}{l} \dot{u}(t)=A u(t) \quad \text { on}\,\,\, (0,T) \\ u(0)...
walid fssm's user avatar
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0 answers
61 views

First isomorphism theorem for inverse semigroups together with v-prehomomorphisms?

In this old paper D. B. McAlister has introduced a very interesting class of morphisms for inverse semigroups, called v-prehomomorphisms. For such a morphism $\theta : S \to T,$ instead of preserving ...
Bumblebee's user avatar
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Minimal Ideals In LA Semigroup

Theorem. For each ideal $I$ of an LA-semigroup $S$, there exists a minimal prime ideal of $I$ in $S$. can any one show the above result for me. I, will be very thankfull.
Junaid Ahmad's user avatar
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A question about v-prehomomorphisms of inverse semigroups

In this paper D. B. McAlister introduced a very interesting class of morphisms for inverse semigroups, which he called v-prehomomorphisms. For a such morphism $\theta : S \to T$ we have $\theta(ab)\...
Bumblebee's user avatar
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4 votes
1 answer
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Do quotient inverse semigroups exist?

A congruence on a semigroup $S$ is an equivalence relation $\sigma\subseteq S\times S$ that respect to the multiplication. In other words $$(a,b), (c,d)\in\sigma \implies (ac, bd)\in\sigma. $$ Given a ...
Bumblebee's user avatar
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Semilattice of the Left Inverse Hull

This is a follow-up on this post, which is based upon this paper. First, let me set up some definitions, etc. A Semigroup $S$ is said to be an inverse semigroup provided that for every $x \in X$, ...
user193319's user avatar
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Inverse Semigroups, Partial Bijections, and Semilattice of Idempotents

I have a question about a passage from this paper. First, some definitions A Semigroup $S$ is said to be an inverse semigroup provided that for every $x \in X$, there exists a unique element $x^{-1}$ ...
user193319's user avatar
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2 votes
0 answers
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Do matrices with multiplication and Moore-Penrose inverse form a regular semigroupoid?

Definition $\left\langle S, \otimes \right\rangle$ is a regular semigroup means The operation $\otimes$ is closed on $S$: $\forall s_1, s_2 \in S: s_1 \otimes s_2 \in S$ The operation $\otimes$ is ...
Charlie's user avatar
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1 answer
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Simple example for a rng with an inverse semi-group as the multiplicative group

I’m looking for a ring without an multiplicative identity, and in which every element $x$ has a weak inverse $y$ such that $xyx=x,yxy=y$ preferably simple to construct and or of finite size. If it has ...
razivo's user avatar
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Characters on inverse semigroups: Hahn-Banach?

Let $S$ be an inverse semigroup with zero $0\in S$, $$ 0s=0=s0,\quad\forall s\in S. $$ Let $e=e^2\in S$ be an idempotent and consider the character, $$ \phi:U\to\{0,1\}:\quad\phi(e)=1,\quad\phi(0)=0, $...
C-star-W-star's user avatar
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2 answers
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example of monoids

An element $x$ of a semigroup $S$ is called regular provided that there exists $y\in S$ such that $xyx=x$. $S$ is called regular if all its elements are regular. Let $S$ be a monoid with identity ...
1Spectre1's user avatar
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2 answers
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Existence of a commutative inverse semigroup with no identity element

Does there exist a commutative inverse semigroup with no identity element, or are all commutative inverse semigroups abelian groups? If there does, what would be an example of such a commutative ...
1103_base_6's user avatar
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0 answers
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Semilattice of idempotent

Let $E$ be a well ordered chain semilattice of idempotent ( $E=\{e_{0}, e_{1}, ... , e_{n}, ... \}$ where $e_{i}\leq e_{j}$ if $i\leq j$). Prove or disprove if $e_{k}e_{h}e_{m}=e_{h}e_{m}$ and $ e_{k}...
Farshad Hasani's user avatar
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1 answer
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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 ...
David Hillman's user avatar
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137 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 $\mathcal J$ ...
user120386's user avatar
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2 votes
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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 ...
Struggler's user avatar
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3 answers
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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: $\...
Momo Tontang's user avatar
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0 answers
67 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.
user avatar
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1 answer
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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 ...
Shaun's user avatar
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7 votes
4 answers
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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 ...
Jakob Elias's user avatar
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2 votes
1 answer
96 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!
Ned Dabby's user avatar
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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 ...
O.morg's user avatar
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1 answer
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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 ...
user403677's user avatar
2 votes
1 answer
188 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 ...
King Ghidorah's user avatar
1 vote
0 answers
123 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 ...
King Ghidorah's user avatar
3 votes
2 answers
86 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 ...
math crab's user avatar
1 vote
2 answers
79 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 ...
guest101018402's user avatar
0 votes
0 answers
71 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{...
bing's user avatar
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5 votes
1 answer
166 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: ...
Mikasa's user avatar
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0 votes
1 answer
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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) $(\...
AvCzar's user avatar
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4 votes
1 answer
221 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 ...
AvCzar's user avatar
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1 vote
0 answers
121 views

Congruence on inverse semigroup

Could you, please, help me 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 ...
Mal JA's user avatar
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0 votes
1 answer
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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)&...
TheCrownedPixel's user avatar
2 votes
1 answer
827 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 ...
KEM's user avatar
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6 votes
1 answer
354 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 ...
Shaun's user avatar
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2 votes
4 answers
1k 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 ...
user100478's user avatar
2 votes
2 answers
212 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 ...
user100478's user avatar
3 votes
1 answer
177 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 ...
goblin GONE's user avatar
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1 vote
1 answer
272 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 ...
Bartek's user avatar
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0 votes
1 answer
106 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 ...
Mr.Lilly's user avatar
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4 votes
1 answer
689 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 $G \cup \{0\}$ ...
Bob's user avatar
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3 votes
0 answers
51 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....
alancalvitti's user avatar
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4 votes
0 answers
145 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 ...
Thomas Klimpel's user avatar
3 votes
1 answer
1k 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 $E_S$ ...
Hassan Muhammad's user avatar