Abstract
An algebra is entropic if its basic operations are homomorphisms. The paper is focused on representations of such algebras. We prove the following theorem: An entropic algebra without constant basic operations which satisfies so called 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. Our theorem is a straightforward generalization of Ježek's and Kepka's theorem for groupoids. As a consequence we obtain that a mode (entropic and idempotent algebra) is a subreduct of a semimodule over a commutative semiring if and only if it satisfies Szendrei identities. This provides a complete solution to the problem in mode theory asking for a characterization of modes which are subreducts of semimodules over commutative semirings. In the second part of the paper we use our theorem to show that each entropic cancellative algebra is a subreduct of a module over a commutative ring. It extends a theorem of Romanowska and Smith about modes.
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