On free modes
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 , Algebra Universalis (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...