Leírás
University of Szeged, Bolyai Institute, Algebra Seminar
Abstract.
Let L be a finite algebraic language with at least one operation symbol of arity >1. By a result of Murskii (1975), a random finite L-algebra is almost surely a semiprimal algebra with no proper subalgebras of size >1. In a recent joint paper with Cliff Bergman (2018+) we looked at the analogous problem when the probability space is restricted to the class of all finite models of a set M of idempotent linear L-identities, i.e., the identities of a strong, idempotent, linear Maltsev condition. We found a simple syntactic condition (*) such that M satisfies (*) if and only if a random finite model of M is almost surely idemprimal.
I will start the talk by reviewing this result, and then I will discuss the following question: Which clones occur with positive probability among the clones of random finite models of M? Clearly, this question is interesting only if (*) fails for M; this is the case, for example, if M is the set of identities for a Maltsev term, or majority term, or minority term, or semiprojection term.