The \(\pi \)-Semisimplicity of Locally Inverse Semigroup Algebras
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In this paper, we first characterize when a semigroup has completely 0-simple semigroup as its principal factors. Let R be a commutative ring with an identity, and let S be a locally inverse semigroup with the set of idempotents locally pseudofinite. Assume that the principal factors of S are all completely 0-simple. Then, we prove that the contracted semigroup algebra \(R_0[S]\) is \(\pi \)-semisimple if and only if the contracted semigroup algebras of all the principal factors of S are \(\pi \)-semisimple. Examples are provided to illustrate that the locally pseudofinite condition on the idempotent set of S cannot be removed. Notice that we extend the corresponding results on finite locally inverse semigroups.
KeywordsLocally inverse semigroup algebras Locally pseudofinite \(\pi \)-Semisimple Completely 0-simple semigroups Principal factors
Mathematics Subject Classification16G30 17C17 17C20 17C27 20M25
- 3.Domanov, A.I.: On semisimplicity and identities of inverse semigroup algebras. Mat. Issled. 38(207), 123–137 (1976) (Russian) Google Scholar