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modes; Szendrei modes; subreducts; semimodules; equational theory
We prove a theorem describing the equational theory of all modes of a fixed type. We use this result to show that a free mode with at least one basic operation of arity at least three, over a set of cardinality at least two, does not satisfy identities selected by 'A. Szendrei in {\it Identities satisfied by convex linear forms\/}, Algebra Universalis {\bf 12} (1981), 103--122, that hold in any subreduct of a semimodule over a commutative semiring. This gives a negative answer to the question raised by A. Romanowska: Is it true that each mode is a subreduct of some semimodule over a commutative semiring?
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