Random models of idempotent linear Maltsev conditions. I. Idemprimality

Abstract

We extend a well-known theorem of Murskiǐ to the probability space of finite models of a system $${\mathcal {M}} $$ of identities of a strong idempotent linear Maltsev condition. We characterize the models of $${\mathcal {M}} $$ in a way that can be easily turned into an algorithm for producing random finite models of $${\mathcal {M}} $$, and we prove that under mild restrictions on $${\mathcal {M}} $$, a random finite model of $${\mathcal {M}} $$ is almost surely idemprimal. This implies that even if such an $${\mathcal {M}} $$ is distinguishable from another idempotent linear Maltsev condition by a finite model $$\mathbf {A}$$ of $${\mathcal {M}} $$, a random search for a finite model $$\mathbf {A}$$ of $${\mathcal {M}} $$ with this property will almost surely fail.