On a quotient S-set induced by countably infinite decreasing chains

Given a monoid S, each S-set may be endowed in a natural way with an Alexandroff topology. Furthermore, if a S-congruence is given on some S-set, we can also endow the corresponding quotient set with an Alexandroff topology. Bearing this in mind, for any monoid S admitting a countably infinite descending chain, we are able to define a specific quotient S-set which is the object of our investigation. More in detail, we carry out a detailed study using several properties of the S-orbits, and we first prove that these quotients are covered by a subcollection of closed subsets related to a suitable notion of dependence on union of subsets. Secondly, we characterize the noetherianity of these quotients in terms of the noetherianity of the monoid S and, finally, we focus our attention to two specific kinds of descending and ascending chains, analyzing some of their main properties on general Alexandroff spaces and, next, showing that the ascending chain stabilizes strictly before than the descending one in the case of our quotients.