On \(\rho\)-semisimple \(S\)-sets (Q2367888)

The author proves theorems of the following kind: If a semigroup \(S\) with 0 is \(P\)-semisimple as an \(S\)-set (\(S\)-act), then every \(S\)-set is \(P\)- semisimple (Th. 3.3). Let \(N_ S \subseteq M_ S\) be an \(S\)-subset and let \(M\) be \(P\)-semisimple, then \(N\) and the Rees factor \(S\)-set \(M/N\) are \(P\)-semisimple. (Th. 3.4). Moreover, \(M\) is \(P\)-finite dimensional iff \(N\) and \(M/N\) are and in this case \(\dim(M) = \dim(N) + \dim(M/N)\) (Th. 3.8). Here \(P\) denotes a special right quotient filter (which is not completely defined in the paper, the missing parts can, for example, be found in [5] of the list of references [the author, ibid. 29, 51-59 (1984; Zbl 0539.20038)]). There, a characterization of a \(P\)-semisimple \(S\)-set \(M\) using the lattice of so-called \(P\)-saturated \(S\)-subsets of \(M\) is given. No examples.