On the semiprimitivity of inverse semigroup algebras and on theorems by Domanov and Munn (Q1824706)

It was shown by \textit{O. I. Domanov} [Mat. Issled. 38, 123-131 (1976; Zbl 0411.20043)] that the semigroup ring K[S] of an inverse semigroup S over a field K is semiprimitive whenever all group rings K[G], G a subgroup of S, are semiprimitive. The converse does not hold in general, but was shown to be true if the lattice E(S) of idempotents of S is a, so called, pseudofinite lattice [\textit{W. D. Munn}, Proc. R. Soc. Edinb., Sect. A 107, 175-196 (1987; Zbl 0627.20041)]. In the paper under review, for a given non-pseudofinite lattice E, the author constructs an inverse semigroup S such that \(E(S)=E\), K[S] is semiprimitive for some field K, and there exists a subgroup G of S such that K[G] is not semiprimitive. This shows that the converse of Domanov's theorem holds for all S with a given lattice of idempotents E, and for all fields K, exactly when E is pseudofinite.