Abstract
A congruence of an algebra is called uniform if all the congruence classes are of the same size. An algebra is called uniform if each of its congruences is uniform. All algebras with a group reduct have this property. We prove that almost every finite uniform Mal’cev algebra with a congruence lattice of height at most two is polynomially equivalent to an expanded group.
Highlights
We say that a congruence θ of an algebra A is uniform if all its congruence classes have the same cardinality
If A is a Mal’cev algebra with congruence lattice isomorphic to Mi for i ≥ 3, A is polynomially equivalent to an expanded group
If A be a finite simple Mal’cev algebra, A is polynomially equivalent to an expanded group
Summary
We investigate finite uniform algebras. We say that a congruence θ of an algebra A is uniform if all its congruence classes have the same cardinality. Key words and phrases: Mal’cev algebra, expanded group, expanded loop, uniform congruence. All previously mentioned classes of algebras: groups, rings, modules, expanded groups, quasigroups, loops, expanded quasigroups, and expanded loops are Mal’cev algebras. All these structures can be seen as expanded quasigroups (loops). The question is: Is there any finite Mal’cev algebra that is uniform and not polynomially equivalent to an expanded quasigroup (loop)?. We prove that each finite uniform Mal’cev algebra with congruence lattice of height at most two is polynomially equivalent to an expanded loop. In many cases, these algebras will be polynomially equivalent to an expanded group
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