3
votes
2answers
153 views

Examples of cancellative normal semigroups

I've got a couple of things to test against normality in cancellative semigroups. A normal semigroup $S$ is one in which for any $x\in S$ we have $xS=Sx.$ This implies the Ore condition $$x,y\in ...
2
votes
2answers
195 views

How much information does the multiplicative semigroup of an algebra contain?

How much do we know about an given algebra when we only know its semigroup strucure under the product law? How far can two algebras be distinguished by knowing only their semigroup strucure? The ...
3
votes
1answer
180 views

Proving that a semigroup is regular

In a number of diverse situations of interest to me (mostly associated with something called the abelian sandpile model), one can define a nonabelian semigroup generated by commuting elements ...
11
votes
1answer
350 views

Automorphisms of $P(\Bbb N)$

I believe I've proved that the power semigroup of non-negative integers with addition has a trivial automorphism group. The proof is a bit long, completely elementary and rather unremarkable (as the ...
1
vote
1answer
124 views

Idempotents in Green J classes

I recently read this article Syntactic semigroups. In page $8$, he speaks about a J class having an idempotent is called regular: A $\mathcal J$-class containing an idempotent is called regular. ...
5
votes
2answers
205 views

For a monoid with zero $M$, how many additive operations on $M$ can there be making $M$ a ring?

Let $(M,\times)$ be a monoid with zero. Let $\Sigma(M,\times)$ be the set of binary operations $+$ on $M$ such that $(M,+,\times)$ is a ring. Let $\sim$ be an equivalence relation on ...
0
votes
1answer
90 views

What are the semigroups in which congruence classes can be multplied like sets?

For a semigroup $S$ and a congruence $\rho$ on $S$, let's say that $\rho$ is good when for all $a,b\in S$ we have that $[ab]=[a][b],$ where $[x]$ denotes the congruence class of $x$ modulo $\rho$ and ...
2
votes
4answers
284 views

On the notion of partial semigroup

A partial binary operation on a set $X$ is just a (partial) function $\varphi: X \times X \rightharpoonup X$ (I'm using \rightharpoonup for partial maps), and a partial magma is a pair $\mathbb M = ...
1
vote
2answers
248 views

What are the monoids in which every globally idempotent subsemigroup contains the identity element?

A semigroup is called globally idempotent when for any $x\in S$ there are $y,z\in S$ such that $x=yz$. Is there a name for monoids whose every globally idempotent subsemigroup contains the identity ...
0
votes
1answer
152 views

When does a power semigroup have a zero, and what can the zero be?

Let $S$ be a semigroup. The power semigroup of $S$ is the set $P(S)=2^S\setminus\lbrace\varnothing\rbrace $ with the operation $$AB=\lbrace ab\ |\ a\in A,\ b\in B\rbrace.$$ This operation is ...
2
votes
1answer
209 views

Embedding Semigroups in Rings

Let $S$ be a finite commutative semigroup with identity. Under what conditions (on the semigroup $S$) it is possible to find a ring $R$ such that the multiplicative structure of $R - \{0\}$ is ...
3
votes
1answer
191 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 ...
6
votes
2answers
331 views
+50

Is the class of inverse semigroups globally determined?

This question is a follow-up to this one I asked on math.stackexchange. I've decided to ask here because I believe this is a research-level question. I'm sorry if I'm wrong -- I'm not a researcher ...
3
votes
2answers
414 views

subsets of $\mathbb{R}^+$ closed under addition

No one's answered this question so here's a simpler question: Has anyone described or catalogued all sets of non-negative real numbers that are closed under addition? In particular, how about those ...
8
votes
3answers
561 views

What is the smallest variety of algebras containing all fields?

A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative inverse semigroup with respect to multiplication. The unique multiplicative ...
8
votes
1answer
411 views

Magma “actions” (or alternatively, “What is the Yoneda lemma for magmas?”)

Arguably the most import thing about groups, semigroups and more generally categories, is that they can act on sets (or even collections of sets in the case of a category). This is the basis for all ...
4
votes
2answers
440 views

Do all finitely generated nilpotent semigroups have polynomial growth?

The notion of nilpotency passes nicely from groups to semigroups. Define $q_1(x,y)=xy$ and $$q_{i+1}(x,y,z_1,\cdots,z_i)=q_i(x,y,z_1,\cdots,z_{i-1})z_iq_i(y,x,z_1,\cdots,z_{i-1})$$ inductively for all ...
1
vote
2answers
323 views

Relations in matrix semigroups

Suppose that $A_1, \dots, A_k \in M_n(\mathbb{Q})$ and $S$ is the semigroup generated by them. Two questions: are there always a finite set of relations $\{R_i\}$ among the $A_j$ such that $S$ is ...