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Embedding entropic algebras into semimodules and modules. (English) Zbl 1193.08001
The paper is focused on representation of entropic algebras. The authors prove the following theorem: An entropic algebra without constant basic operations which satisfies Szendrei identities and such that all its basic operations of arity at least two are surjective, is a subreduct of a semimodule over a commutative semiring. Their theorem is a generalization of Ježek and Kepka’s theorem for groupoids. They obtain that a mode is a subreduct of a semimodule over a commutative semiring if and only if it satisfies Szendrei identities. This provides a complete solution to a problem in mode theory. In the second part of the paper, the authors use their theorem to show that each entropic cancellative algebra is a subreduct of a module over a commutative ring. This extends a theorem of Romanowska and Smith about modes.

MSC:
08A05 Structure theory of algebraic structures
16Y60 Semirings
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