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Generalized entropy in expanded semigroups and in algebras with neutral element

Lehtonen, Erkko; Pilitowska, Agata

Semigroup Forum, 88(3) (2014), 702–714

The complex product of (non-empty) subalgebras of a given algebra from a variety V is again a subalgebra if and only if the variety V has the so-called generalized entropic property. This paper is devoted to algebras with a neutral element or with a semigroup operation. We investigate relationships between the generalized entropic property and the commutativity of the fundamental operations of the algebra. In particular, we characterize the algebras with a neutral element that have the generalized entropic property. Furthermore, we show that, similarly as for n-monoids and n-groups, for inverse semigroups, the generalized entropic property is equivalent to commutativity.