# 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.

589 questions
Filter by
Sorted by
Tagged with
134 views

### Centers and conjugacy classes of groups relative to a pair of group homomorphisms

$\newcommand{\defeq}{\mathbin{\overset{\mathrm{def}}{=}}}$Given a group $G$, its center $\mathrm{Z}(G)$ and set of conjugacy classes $\mathrm{Cl}(G)$ are defined by \begin{align*} \mathrm{Z}(G) &\...
• 11.3k
117 views

### On dual notions of morphisms of algebraic structures obtained by replacing equaliser with coequalisers

This question is based on this discussion from the Category Theory Zulip. See also the earlier question Natural cotransformations and "dual" co/limits. Let $G$ and $H$ be groups. We define ...
• 11.3k
399 views

165 views

• 26.2k
67 views

### Structure of well-ordered commutative monoids

Let $(M,+)$ be a commutative monoid. Let $<$ be a well-ordering on $M$, where $\forall a\in M,\ 0\leq a$ $\forall a,b,c\in M,\ a<b\Rightarrow a+c<b+c$ The first condition means $M$ will be ...
• 18.2k
74 views

### An alternative definition for finitely generated (and principal) ideals in a semigroup

Let $S$ be a semigroup. An ideal (of $S$) is a subset $I$ of $S$ such that $SI$ and $IS$ are both contained in $I$. The non-empty ideals constitute a subsemigroup, $\mathfrak I(S)$, of the power ...
• 9,646
581 views

• 9,646
### Prove or disprove: $R^{n+1} \supseteq R \cap R^2 \cap \cdots \cap R^n$ for every binary relation $R$ on a set of size $n$
Prove or disprove: $R^{n+1} \supseteq R \cap R^2 \cap \cdots \cap R^n$ for every binary relation $R$ on a set of size $n$. I have verified the statement for $n \leq 4$ with a Mathematica code. I have ...