# 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.

494 questions
Filter by
Sorted by
Tagged with
60 views

### Nonzero idempotents in compact semitopological semigroups with zero

Let $S$ be a compact semitopological semigroup. Then, $S$ contains minimal idempotents by Ellis' theorem. Ellis' Theorem: In a compact left-topological (resp. right-topological) semigroup, every ...
135 views

### Matrix decompositions as monoid isomorphisms. Ever considered before?

I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question: ...
105 views

### Semigroup ideals of a ring or an algebra

Let $R$ be a ring or an algebra. Suppose $B\subseteq R$ satisfies the property $BR \subseteq B$ and $RB\subseteq B$. Is there a general theory of subsets with this property in a ring (resp. algebra) ...
80 views

### The "matrix direct sum" monoid modulo unitary equivalence

Given a commutative $*$-ring $(R,*)$, let $M(R,*)$ be the monoid whose elements are matrices over $R$ of all possible shapes and entries, including those that have $0$ columns or $0$ rows. Let the ...
145 views

139 views

### Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?

It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G. More generally here (MO275769) Qiaochu Yuan ...
88 views

### Submonoid of free monoid with certain properties

Let $N$ be a submonoid of a free monoid $M$ such that $m_1nm_2\in N \Rightarrow m_1,m_2\in N$ for any $m_1,m_2\in M$ and $n\in N\setminus\{1\}$. $\quad\quad\quad\quad$ (C) Do such submonoids ...
262 views

### What is the precise connection between logarithmic algebraic geometry and the field with one element?

Monoid schemes (a.k.a. $\frak M$-schemes) have been introduced by Deitmar as a possible approach to geometry over the field with one element. These build upon monoids as the basic building blocks for ...
56 views

92 views

### Partial orders on $\mathbb{N}^m$ compatible with addition

I'm looking for a classification (or just non-trivial examples) of partial orders on monoid $\mathbb{N}^{m}$ that are compatible with addition. That is, partial orders $\leq$ satisfying two additional ...
Given a semigroup $(S, *)$ we extend the semigroup operation $*$ of $S$ to a operation $*$ on $\beta S$ (the set of ultrafilters on $S$) defined as $$\mathcal{U} * \mathcal{V} = \left\{ A \... 1answer 128 views ### Cohomology of commutative monoid acting on module I have a some naive questions about how to define the cohomology of a commutative monoid. One way to express the cohomology of a group G with coefficients in a module A is as \text{Ext}^i_{\... 1answer 299 views ### Are there overwhelmingly more finite monoids than finite spaces? [closed] A function f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1} overwhelms g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1} if for any k\in \mathbb{Z}_{\geq 1} the inequality f(n)\leq g(n+k) holds only for ... 0answers 65 views ### Terminology and notation for generated subgroups I would like to think about formation of the smallest subgroup (or monoid, or whatever) H of G containing two given subgroups A and B as an operation on subgroups, and I wonder if there is a ... 0answers 103 views ### Free monoids on posets I've suddenly found myself working with some free monoids F(S) in which the set S is a poset, and the order extends to an order F(S), satisfying if (but not only if) s_1, s_2, \ldots, s_r, t_1, ... 0answers 62 views ### When is the submonoid preserving a subspace finitely generated? Let T be a topological space with at least one open set whose closure is not open. Let G be a finitely generated group acting by homeomorphisms on T. Let S\subset T be a subspace. Under what ... 1answer 453 views ### Is the Petersen graph a "Cayley graph" of some more general group-like structure? The Petersen graph is the smallest vertex-transitive graph which is not a Cayley graph. Is it the "Cayley graph" of some slightly more general group-like structure? 1answer 255 views ### Lax monoidal functor Let me denote Cat the category of small categories. It is a symmetric monoidal category with respect to the cartesian product. Let F: (Cat, \times)\rightarrow (Set,\times) a symmetric monoidal ... 1answer 661 views ### Are there any "simple" monoids with intermediate growth? The discovery of the Grigorchuk group which has intermediate growth caused a number of other such groups to be found, but they are all fairly complicated, and as far as I know none of them are ... 0answers 62 views ### Size of the kernel (minimal ideal) of a finite semigroup Let A be an irreducible nonnegative N\times N integer matrix with constant row sum D. Let A_1, \dots, A_D be nonnegative integer matrices, each with constant row sum 1, such that \sum_k A_k ... 0answers 159 views ### The forgetful functor from Groups to Semigroups While teaching this term I found myself reminded of the fact that the "usual" definition of a group homomorphism is really the definition of a semigroup homomorphism, applied to semigroups ... 0answers 136 views ### Is the group ring of an amenable group, viewed as multiplicative monoid, amenable? Motivated by this question, it seems natural to ask the following: Question 1: Is there a [finitely generated discrete] torsion-free virtually Abelian (but not Abelian) group G so that the ... 1answer 79 views ### Derivable relations in a monoid Let  X  be a monoid which is generated by the elements  x_1, x_2, \hat x_1, \hat x_2  and the relations  \hat x_i x_i = 1  and  x_i \hat x_j = \hat x_j x_i  for any distinct  i, j = 1, 2 . ... 0answers 70 views ### Quotient of monoids and monoid algebras Let  X  be a monoid and  R  be a (two-sided) congruence relation on  X  which is generated by some relations  u_i \equiv_R v_i  for any  i  in some index set  J . Let  K  be a field,  K[... 0answers 98 views ### Cuntz semigroups of basic C*-algebras I am doing some research related to Cuntz semigroups, and I am trying to find concrete examples in simple cases. In one paper that I found, it says the following (p.103): "[...] A_i is ... 0answers 145 views ### Do you know rings without involutions, auto-anti-isomorphics? In that case, what is the minimal example? Do you know rings without involutions, but auto-anti-isomorphic (isomorphic to their opposite)? In that case, what is the minimal example? If a ring has an involution f, then f is an anti-automorphism;... 0answers 63 views ### Monoids with three or more "natural" partial orders For any given monoid M there may exist lots and lots of compatible pre-orders \leq. Only few of these are usually any interesting though. I can find some examples of monoids that have two non-... 1answer 115 views ### Well-foundedness of divisibility vs well-foundedness of right- and left-divisibility Say that a preorder (i.e., a reflexive and transitive binary relation) \preceq on a set X is artinian if there is no sequence (x_n)_{n \ge 1} of elements of X with x_{n+1} \prec x_n for ... 1answer 115 views ### Is it true that the structure of a commutative ordered semiring is unique on a commutative ordered monoid? Is it true that the structure of a commutative ordered semiring with identity is unique on a commutative ordered monoid (i.e., the structure of the monoid and the order are consistent)? I am not ... 1answer 314 views ### Computations of divisor class monoids Let me first recall some definitions from the very first pages of Bourbaki, Commutative Algebra, Chapter 7, "Divisors". Let A be a (commutative) domain, K its field of fractions. A ... 1answer 124 views ### Reference request: a cousin to the log semiring Let f be strictly increasing on \mathbb{R}. Then x \oplus y := f^{-1}(f(x)+f(y)) gives rise to a strict symmetric monoidal (\Rightarrow commutative monoid) structure on (\mathbb{R},\ge) with ... 3answers 2k views ### Is each squared finite group trivial? A semigroup S is defined to be squared if there exists a subset A\subseteq S such that the function A\times A\to S, (x,y)\mapsto xy, is bijective. Problem: Is each squared finite group ... 0answers 79 views ### Flag variety as monoid and Schubert calculus The lattice of linear subspaces in a vector space V can be provided with a structure of monoid by considering the subspace generated by the union of two subspaces as the monoid operation. When looking ... 0answers 70 views ### Cayley's theorem for regular semigroups "Cayley's theorem" for semigroups says that every semigroup of size n is isomorphic to a subsemigroup of the semigroup of transformations T_n or T_{n+1}. For inverse semigroups, we ... 1answer 163 views ### Conjugacy classes of monoids II: Abelianising a monoid, wrongly \newcommand{\unsim}{\mathord{\sim}}Let G be a group. What is$$ G/\left(ab\sim ba\ \middle|\ a,b\in G\right)?  Answer: not $G^{\mathrm{ab}}$, but the set of conjugacy classes of $G$. When ...
It is the monoid obtained by adjoining all solutions of polynomial equations. I'll demonstrate how to adjoin a single solution to a polynomial equation to a monoid: Let $W$ be a monoid and let \$p(x)=q(...