Generalized entropy in expanded semigroups and in algebras with neutral element |
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Authors: | Erkko Lehtonen Agata Pilitowska |
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Institution: | 1. Computer Science and Communications Research Unit, University of Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359, Luxembourg, Luxembourg 2. Centro de álgebra da Universidade de Lisboa, Avenida Professor Gama Pinto 2, 1649-003, Lisbon, Portugal 3. Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662, Warsaw, Poland
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Abstract: | The complex product of (non-empty) subalgebras of a given algebra from a variety \(\mathcal{V}\) is again a subalgebra if and only if the variety \(\mathcal{V}\) 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. |
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