## Micha\l Marek Stronkowski

*On free modes *

Comment.Math.Univ.Carolin. 47,4 (2006) 561-568. **Abstract:**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 {Identities satisfied by convex linear forms}, Algebra Universalis {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?

**Keywords:** modes, Szendrei modes, subreducts, semimodules, equational theory

**AMS Subject Classification:** 08B20, 03C05, 03F07

PDF