On Polynomials in Mal’cev Algebras

Abstract

<p>We establish several properties of higher commutators, which were<br />introduced by A. Bulatov, in congruence permutable varieties. We use these<br />commutators to prove that the clone of polynomial functions of a finite Mal&rsquo;cev<br />algebra whose congruence lattice is of height at most 2, can be described by a<br />finite set of relations. For a finite nilpotent algebra of finite type that is a product<br />of algebras of prime power order and generates congruence modular variety, we<br />are able to show that the property of affine completeness is decidable. Moreover,<br />polynomial equivalence problem has polynomial complexity in the length of the<br />input polynomials.</p>

<p>Ustanovljavamo osobine viˇsih komutatora, koje je uveo A. Bulatov,<br />u kongruencijki permutabilnim varijetetima. Te komutatore koristimo da bi<br />dokazali da se klon polinomijalnih funkcija konaˇcne Maljcevljeve algebre ˇcija je<br />mreˇza kongruencija visine najviˇse dva moˇze opisati konaˇcnim skupom relacija. Za<br />konaˇcne nilpotentne algebre konaˇcnog tipa koje su proizvod algebri koje imaju red<br />stepena prostog broja i koje generiˇsu kongruencijki modularan varijetet pokazu-jemo da je osobina afine kompletnosti odluˇciva. Takod&macr;e, pokazujemo za istu klasu<br />da problem polinomijalne ekvivalencije ima polinomnu sloˇzenost u zavisnosti od<br />duˇzine unetih polinomijalnih terma.</p>