0
votes
1answer
52 views

If $G,H$ are monoids and not groups, prove that $f(e_G)=e_H$ may be wrong.

Problem: If $f:G\longrightarrow H$ is a homomorphism of groups, then $f(e_G)=e_H$ and $f(a^{-1})=f(a)^{-1}, \forall a\in G$. Show by example that the first conclusion may be false if $G, H$ are ...
2
votes
2answers
29 views

Semigroups and units

I am working my way though Basic Algebra 1. I am currently on chapter 1 , and more specifically I busy with the following exercise: "Let $S$ be a set a define a product in $S$ by $ab=b $. Show that ...
7
votes
2answers
92 views

Are these adjoint functors to/from the category of monoids with semigroup homomorphisms?

Do the forgetful functors $G_H:\bf Monoid \to \bf Semigroup^1$ and $G_O:\bf Semigroup^1 \to \bf Semigroup$ have left and/or right adjoints? Here $\bf Semigroup$ is the category of semigroups with ...
1
vote
1answer
38 views

positive integer polynomial under the usual polynomial multiplication

consider the set of polynomials with positive integer coefficients together with the operation, usual multiplication of polynomials. now my first question is does this set together with the ...
1
vote
1answer
30 views

Findout if $\min(x,y)$ and $\max(x,y)$ are semigroup, monoid or group

I need to find out whether $A \oplus B := \max(A,B)$ or $A \ominus B := \min(A,B)$ form a semigroup, monoid or group for $\oplus : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ and $\ominus : ...
1
vote
1answer
53 views

In a dagger category, what do we call an arrow $f$ such that $f \circ f^\dagger \circ f = f$?

In a dagger category, what do we call an arrow $f$ satisfying $f \circ f^\dagger \circ f = f$? In $\mathrm{Rel}$ (and, more generally, an allegory) this is straightforwardly equivalent to $f \circ ...
1
vote
1answer
46 views

Generalising cover maps from monoids to semigroups

Let $T,S$ be monoids. A partial surjective mapping $\psi : T \to S$ is called a cover map if for each $s \in S$ there exists some $\hat{s} \in T$ called a cover of $s$ such that for each $t \in ...
3
votes
2answers
74 views

On the powerset of a monoid, does the second operation play nice with the first? Indeed, is it even useful?

Let $X$ denote a monoid. Then we can make $Y = \mathcal{P}(X)$ into a monoid, too. Define $$AB = \{ab \mid a \in A, b \in B\}$$ for all $A,B \in Y.$ We see immediately that $1$ (shorthand for $\{1\}$) ...
4
votes
1answer
86 views

Maximal ideals in rings

Let $R$ be a ring with identity. Is it true that if $R$ has a finite maximal right ideal then it MUST have a finite maximal two-sided ideal ?
2
votes
1answer
81 views

Equational theories in commutative idempotent monoids

It might well be that my question is trivial but I'm not a mathematician, I just need a formalization for algebraic semiotics. If I have a commutative idempotent monoid, I can define a partial ...
4
votes
1answer
31 views

Semigroup which satisfied $(xy)^{\pi} = (xy)^{\pi}x$ and principal right ideals

Let $(S,\cdot)$ be a finite semigroup, then every $s \in S$ has an unique idempotent power, i.e. there exists a smallest $i \in \mathbb N$ such that $s^i$ is idempotent. It is the unique idempotent in ...
2
votes
1answer
181 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 $G \cup \{0\}$ ...
2
votes
2answers
130 views

What should I call this commutative monoid of order three?

I'm looking for a name for the monoid given by the following table: $$ \begin{array}{c|ccc}&1&a&b\\ \hline 1&1&a&b\\ a&a&1&b\\ b&b&b&b \end{array} $$ ...
4
votes
1answer
125 views

If $z$ is the unique element of a monoid such that $uzu=u$, is $u$ invertible?

This question is a follow-up to this one. I tried to check whether the same statement as discussed for rings there is true for monoids too, but without success. Let $M$ be a monoid and $u\in M$. ...
3
votes
1answer
62 views

What condition on $m$ turns this semigroup into a monoid?

Suppose you have a monoid $(M,p,1)$ (viewing it as a triple of a set $M$, operation $p$, and unit $1$). Then for some $m\in M$ we can define a new product $p_m$ in $M$ by $p_m(a,b)=amb$. It's easy to ...
8
votes
3answers
261 views

give a counterexample of monoid

$G$ is a monoid,$e$ is its identity,if $ab=e$ and $ac=e$, can you give me a counterexample such that $b\neq c$. if not,please prove $b=c$ thanks a lot.
1
vote
1answer
118 views

Does a binoid contain an empty word?

$A$ is a finite alphabet. $A^*$ is the set of finite words or the free monoid generated by A. $A^w$ is the set of infinite words generated by A. Denote $A^\infty=A^*\bigcup A^w$. $X$ is a set of ...
26
votes
3answers
928 views

What can we learn about a group by studying its monoid of subsets?

If $G$ is a group, then $M(G)=2^G$ is has a monoid structure when we define $AB$ to be $\{ab|a\in A,b\in B\}$ and $1_{M(G)}=\{1\}$. How much of the structure of $G$ can be recovered by studying the ...
2
votes
0answers
150 views

Constructing a semigroup from a small category

The following was given as an example for a semigroup without an identity: Finite sets of matrices of varying dimensions, where the product A*B={PQ|P in A & Q in B & dim(Q)=codim(P)}, and ...
6
votes
1answer
554 views

Why is the free monoid free?

I have been reading up on monoids recently and came across the free monoid Σ*, which (if I understand correctly) is an initial object in the category of monoids, meaning that there is a ...
21
votes
9answers
1k views

Are there any interesting semigroups that aren't monoids?

Are there any interesting and natural examples of semigroups that are not monoids (that is, they don't have an identity element)? To be a bit more precise, I guess I should ask if there any ...