Suppose $S$ is a semigroup (or a monoid, or a category), and $X$ is an $S$-set -- that is, a set with an action of $S$. Say that a sub-$S$-set $Y$ is "downward closed" (or maybe "well-generated") if for any $x \in X$ such that $sx \in Y$ for some $s \in S$, we have $x \in Y$.

Has this notion been defined before? I'm really mostly wondering because I'd like to give this a good name.

Notice that if $S$ is a group, this just collapses to any sub-$S$-set since we have inverses. On the other hand, the analogous notion for a module would be the following: a submodule $N$ of $M$ is downward closed if whenever $rm \in N$, $m \in N$. This is related to the notion of a dense submodule, but distinct.

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    $\begingroup$ Saturated maybe? $\endgroup$ – Gabriel C. Drummond-Cole Oct 8 '15 at 4:25
  • $\begingroup$ @GabrielC.Drummond-Cole Great! That looks like the right term at least for modules. It also looks like it's at least used for the action of $\mathbb{N}$ on a semigroup. I can't find any references for its use in the general context of semigroup actions, but absent other names I think I'll go with that. $\endgroup$ – Kevin Casto Oct 8 '15 at 4:54
  • $\begingroup$ If I understand what you mean, the analogous notion for modules is degenerate because $0\cdot m$ is in $N$ for every $m\in M$. $\endgroup$ – Julian Rosen Oct 8 '15 at 17:13
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    $\begingroup$ The mention of saturated reminds me of the question blip that was MO 126742... $\endgroup$ – Benjamin Dickman Oct 8 '15 at 23:58
  • $\begingroup$ @JulianRosen Sorry, I was too brief in my answer: the right analogue for a module should be only for $r$ a regular element, i.e. not a zero-divisor. So an equivalent definition would be that $M/N$ is torsion-free. This is apparently what people call a saturated submodule. $\endgroup$ – Kevin Casto Oct 9 '15 at 19:14

People have thought about it. I don't know if it has a name. If S is a semigroup acting on a set X, then consider the least equivalence relation on X such that x is equivalent to sx for all x in X and s in S. In my paper http://www.combinatorics.org/ojs/index.php/eljc/article/download/v17i1r164/pdf I call the equivalence classes for this relations the weak orbits of S on X. Clearly your closed sets are precisely those which are unions of weak orbits.


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