2
$\begingroup$

Definition. A semigroup $X$ is called $E$-separated if for any distinct idempotents $x,y\in X$ there exists a homomorphism $h:X\to Y$ to a semilattice $Y$ such that $h(x)\ne h(y)$.

Observe that $X$ is $E$-separated if and only if the smallest semilattice congruence on $X$ is idempotent-separating. This seems to be an important notion, so I suggest that it could (and should) be studied in Theory of Semigroups, maybe under some different name. Do you know any suitable references?

Improve this question
$\endgroup$
3
  • $\begingroup$ For finite semigroups this is equivalent to every regular J-class being a group and the class is known as DG (D-classes are groups). I always forget the description for infinite semigroups of the maximal semilattice components but I'm sure your class is something like a semilattice of nilpotent extensions of some nicer thing with a unique idempotent plus of course all semigroups with no idempotents $\endgroup$
    – Benjamin Steinberg
    Commented Feb 11, 2022 at 11:42
  • $\begingroup$ @BenjaminSteinberg Yes, you are right. The description is something like that you wrote. What I need is to name somehow such semigroups (that admit this nice description). I would prefer to call them $E$-separated, especially if nothing different exists sofar. $\endgroup$
    – Taras Banakh
    Commented Feb 11, 2022 at 12:19
  • 1
    $\begingroup$ I'm not aware of any name beyond the finite case where DG is the official terminology $\endgroup$
    – Benjamin Steinberg
    Commented Feb 11, 2022 at 13:12

1 Answer 1

Reset to default
1
$\begingroup$

I finally found an answer to my own question: by an old (nontrivial) result of Putcha and Weissglass, a semigroup $X$ is $E$-separated if and only if it is viable.

A semigroup $X$ is viable if for any elements $x,y\in X$ with $\{xy,yx\}\subseteq E(X)$ we have $xy=yx$.

However, up to my taste, the notion of viality is a bit less intuitive comparing to the equivalent notion of $E$-separatedness.

Another difference between those (equivalent) notions is that the viality is an internal property whereas the $E$-separatedness is external (defined with the help of external objects). But the $E$-separatedness also can be equivalently defined in internal terms: a semigroup is $E$-separated if the smallest semilattice congruence on $X$ separates idempotents.

Neither $E$-separatedness not the viablity are present in this Wikipedia page. However they form an important class of semigroups.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .