# Random Models of Idempotent Linear Maltsev Conditions. I. Idemprimality

## TL;DR

This paper extends a theorem to characterize and generate random finite models of certain algebraic identities, showing that such models are almost surely idemprimal, impacting the approach to model search.

## Contribution

It introduces an algorithm for producing random finite models of strong idempotent linear Maltsev conditions and proves their almost sure idemprimality under mild restrictions.

## Key findings

- Random models are almost surely idemprimal.
- Finite model search for distinguishing conditions is almost surely ineffective.
- Extension of Murski theorem to probabilistic models.

## Abstract

We extend a well-known theorem of Murski\v{\i} 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.