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Some applications of higher commutators in Mal’cev algebras. (English) Zbl 1206.08003

Many properties of higher commutators in congruence-permutable varieties are derived. Let \(A\) be a finite nilpotent algebra of finite type that is a product of algebras of prime power order and generates a congruence-modular variety. Using higher commutators, the following results are proved: There is an algorithm that decides whether every congruence-preserving function on \(A\) is a polynomial function on \(A\) or not. The problem of deciding if two given polynomial terms of \(A\) induce the same function has polynomial time complexity in the length of the input terms.

MSC:

08A40 Operations and polynomials in algebraic structures, primal algebras
08B10 Congruence modularity, congruence distributivity
68Q25 Analysis of algorithms and problem complexity
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