6
$\begingroup$

Let $\mathscr S$ be the class of all semigroups with zero. For $(S,\times,0)\in\mathscr S,$ I want to count additive operations $+$ on $S$ such that $(S,+,\times,0)$ is a ring (possibly without unity). For $S\in\mathscr S,$ let $\Sigma_S$ be the set of all such additive operations on $S$. Let, for $+_1,+_2\in \Sigma_S,$ define $+_1\sim +_2$ to mean that $(S,+_1,\times,0)\cong (S,+_2,\times,0).$

Let $\Sigma'_S:=\Sigma_S/\sim.$

Let $\kappa_S:=\operatorname{card}(\Sigma_S)$ and $\kappa'_S:=\operatorname{card}(\Sigma'_S).$

Are there upper bounds to the values of $\kappa_S$ and $\kappa'_S$ for $S\in \mathscr S?$ Do they admit all non-negative integer values? If not, which non-negative integer values do they admit?

These questions are not similar to anything I know and I don't even know how to search for answers in literature. I'm especially curious what the behavior of $\kappa_S$ and $\kappa'_S$ is for finite semigroups $S.$

EDIT I have found this question on MO. Arturo Magidin gives an example there of two non-isomorphic rings having isomorphic multiplicative structures. If I understand correctly, this is proven by using the unique factorization property in polynomial rings over rings with the unique factorization property.

So my question isn't trivial, which I didn't know at the time of asking: there is a semigroup with zero for which there are at least two additive operations making the semigroup two non-isomorphic rings.

EDIT An answer to the following questions wouldn't be off-topic:

Is there a semigroup $S$ with $0$ such that $\kappa_S$ is infinite? Is there a semigroup $S$ with $0$ such that $\kappa'_S$ is infinite?

(If the answer is "yes", this doesn't answer any of my previous questions, but I would be interested in knowing it nonetheless.)

  • 0
    It's possible that mathoverflow might be a better place for this question?2012-03-14

0 Answers 0