# Questions tagged [semigroups-and-monoids]

A semigroup is a set $S$ together with a binary operation that is associative. Examples of semigroups are the set of finite strings over a fixed alphabet (under concatenation) and the positive integers (under addition, maximum, or minimum). A monoid is a semigroup with a neutral element. Of course, any group is also a monoid/semigroup.

422 questions
Filter by
Sorted by
Tagged with
100 views

### Lie monoids as monoids internal to the category of smooth manifolds?

This question can be thought as a complement to this one. Lie groups can be defined as groups internal to the category of smooth manifolds. Lie monoids, however, as a particular case of Lie semigroups,...
23 views

### Looking for a characterization of 'P2-monoids' and related structures

Definition 1. Let $M$ denote a monoid and consider $x \in M$. Then for the purposes of this question, $x$ is semicentral if and only if for all $a,b \in M,$ with $ba = 1$, we have $bxa=x$. Some basic ...
146 views

### Surjective monoid homomorphism $\text{End}(B)\to \text{End}(A)$ given surjection $g:B\to A$

For any set $A\neq\varnothing$ let $\text{End}(A)$ denote the endomorphism monoid, consisting of all functions $f:A\to A$, together with composition. If $A, B\neq \varnothing$ are sets and $g:B\to A$ ...
165 views

### Can one turn finite-dimensional vector subspaces into a cancellative semigroup?

Let $V$ be a vector space over some field and let ${\rm Fin}\,V$ be the family of all finite-dimensional subspaces of $V$. Is it possible to turn ${\rm Fin}\,V$ into an commutative cancellative ...
67 views

### Non-commutative version of the order dimension of a poset

I view the order dimension of a poset $P$ as an inherently commutative notion. On the one hand, it can be defined via realizers, which I find fairly intuitive from an order-theoretic viewpoint. On the ...
146 views

590 views

109 views

74 views

### Ascending sequences of idempotents in inverse semigroups

I've enocuntered the following question in my current research, and I'd appreciate any help you could give me. This is probably well known to experts on the subject. Let $S = \langle K \rangle$ be a ...
22 views

203 views

### What is a “cusp” (“кусок”) in relation to Guba's embedding theorem?

I'm confused by the definition of a "cusp" as found in V.S. Guba, Conditions for the embeddability of semigroups into groups, Math. Notes 56 (1994), Nos. 1-2, 763-769 (link). In the words of Mark ...
74 views

### Is the natural action of the monoid of endomorphisms is a complete invariant for group?

Let $\alpha$ and $\beta$ be actions of semigroups $A$ and $B$ on sets $X$ and $Y$ respectively. Recall that $\alpha$ and $\beta$ are called isomorphic if there exists an isomorphism $\phi$ between ...
106 views

### Classification of associative polynomial functions

What is known about a classification of associative (binary) polynomial functions? First of all, it is interesting in two cases: over Integral domain (or even over field) and over ring of integers ...
615 views

### Associativity may fail by little?

It is a well-known result on group theory that if a group has many pairs of commuting elements then it is abelian. This motivated the following pseudo-conjecture. If a (possibly infinite) set $S$ ...
297 views

145 views

### Comonoids in the category of monoids

Let us give the category of monoids $\mathbf{Mon}$ a monoidal structure with $\otimes = \sqcup$ (coproduct). How can we classify $\mathbf{CoMon}(\mathbf{Mon})$, the category of comonoids of monoids? ...
37 views

### Compute irreducibles of monoid

Given $n > 0$ and $w \in \mathbb{Z}^n$. Is there an efficient algorithm to compute the set of irreducible elements of the monoid $M_w = \{x \in \mathbb{N}^n \mid \langle x,w\rangle = 0 \}$? Here, ...
177 views

### Continuous semigroup homomorphism of composition to additive structure

Let $G$ be the topological semigroup whose underlying space is $C(\mathbb{R}^d,\mathbb{R}^d)$ equipped with composition as semigroup operation and let $H$ be the topological group whose underlying ...
955 views

### Monoids of endomorphisms of nonisomorphic groups

Can monoids of endomorphisms of nonisomorphic groups be isomorphic ?
111 views

### Internal commutative monoid gives commutative monad

Let $(C,\otimes,1)$ be a symmetric monoidal category. Let $(M,\mu,\eta)$ be an internal commutative monoid object. The functor $X\mapsto M\otimes X$ has a canonical monad structure, with unit and ...
89 views

### Joint spectral radius of $\{M,M^T\}$

Let $F$ be a bounded subset of ${\bf M}_n({\mathbb C})$. G.-C. Rota & G. Strang defined the joint spectral radius of $F$ as follows. For $k\ge1$, denote $F_k$ the set of all products of $k$ ...
80 views

### Cancellativity of a particular $2$-generated monoid presented by an infinite number of relations

Let $X = \{x, y\}$ be a two-element set, and let $H$ be the monoid defined by the presentation $$\langle x, y \mid x y^k x = y x y^{k+1} x y, \text{ for } k = 0, 1, 2, \ldots\rangle.$$ That is, $H$ ...
52 views

### Are there finitely-presented astral monoids?

We say a semigroup $S$ is $k$-astral if there exists a finite set $F \subset S$ such that whenever $s_1, s_2, ..., s_k \in S$ there exists $s \in S$ such that $\forall i: s_i \in sF$. Say $S$ is ...
31 views

### $m$-potent elements in $N(C_n)$

We konw that in semigroup theory, $(C_n)$ is the semigroup of all order-preserving and decreasing transformations on $X_n=\{1,\dots,n\}$, under its natural order and $N(C_n)$ the set of all nilpotent ...
170 views

### Generating the monoid of injective endomorphisms of the free group

Let $F$ be the free group of rank $2$ (or any finite rank if this does not matter). The set of injective group endomorphisms $F\to F$ forms a monoid $M$ by compositions. Is there a simple looking set ...
85 views

96 views

### Generalization of pseudogroups

Pseudogroups are defined here: https://ncatlab.org/nlab/show/pseudogroup One of the problems with defining manifolds in terms of pseudogroups is that it gives no notion of a morphism between manifolds,...
511 views

### Is the equational theory of groups axiomatized by the associative law?

Consider the class of groups in the signature {*}. Is the equational theory of that class axiomatized by the associative law? I asked this on math stack exchange but I didn't receive a satisfactory ...
139 views

### A characterisation of faces of rational polyhedral cones

This is about a (seemingly) basic lemma about rational polyhedral cones that is sometimes used when working with toric varieties and is usually "left to the reader". Unfortunately, I could ...
139 views

### Which homotopy types can be realized as the classifying space of a right-cancellative discrete monoid?

McDuff showed that every connected homotopy type can be realized as the classifying space of a discrete monoid, but the monoid she constructs has lots of idempotents. Question: Which homotopy types ...
1k views

### Who invented Monoid?

I was trying to find (and failed) the original author of either the concept of Monoid (set with binary associative operation and identity) the name (which sounds french ? and also Dioid (for what ...
170 views

### Topological category of topological monoids / operads

The category of topological monoids can be made into a topological category in a naive way. Namely, the space of all continuous homomorphisms between two topological monoids is a closed subspace of ...
In this question, it is shown that all Archimedean ordered groups are isomorphic to an ordered subgroup of $\mathbb R$. Additionally, it is shown that if such a group is complete, then it is ...
Consider the category $\mathrm{TopMon}$ of topological monoids and continuous monoid homomorphisms. Consider the inclusion $i:\Bbb{R}_{\ge 0}\hookrightarrow \Bbb{R}$, where the spaces are taken with ...