Csaba Szabó

THE COMPLEXITY OF CHECKING IDENTITIES OVER FINITE GROUPS

We analyze the computational complexity of solving a single equation and checking identities over finite meta-abelian groups. Among others we answer a question of Goldmann and Russel from 1998: we prove that it is decidable in polynomial time whether or not an equation over the six-element group S3 has a solution.

Order Varieties and Monotone Retractions of Finite Posets

In this paper we introduce a new version of the concept of order varieties. Namely, in addition to closure under retracts and products we require that the class of posets should be closed under taking idempotent subalgebras. As an application we prove that the variety generated by an order-primal algebra on a finite connected poset P is congruence modular if and only if every idempotent subalgebra of P is connected. We give a polynomial time algorithm to decide whether or not a variety generated by an order-primal algebra admits a near unanimity function and so we answer a problem of Larose and Zádori.

The complexity of the word-problem for finite matrix rings

We analyze the so-called word-problem for M 2 (Z 2 ), the ring of 2 × 2 matrices over Z 2 . We prove that the term-equivalence problem for the semigroup (and so for the ring) M 2 (Z 2 ) is coNP-complete.

A NEW OPERATION ON PARTIALLY ORDERED SETS

Abstract Recently it has been shown that all non-trivial closed permutation groups containing the automorphism group of the random poset are generated by two types of permutations: the first type are permutations turning the order upside down, and the second type are permutations induced by so-called rotations. In this paper we introduce rotations for finite posets, which can be seen as the poset counterpart of Seidel-switch for finite graphs. We analyze some of their combinatorial properties, and investigate in particular the question of when two finite posets are rotation-equivalent. We moreover give an explicit combinatorial construction of a rotation of the random poset whose image is again isomorphic to the random poset. As a corollary of our results on rotations of finite posets, we obtain that the group of rotating permutations of the random poset is the automorphism group of a homogeneous structure in a finite language.

ON THE FREE SPECTRUM OF THE VARIETY GENERATED BY THE COMBINATORIAL COMPLETELY 0-SIMPLE SEMIGROUPS

We give an asymptotic bound for the size of the n -generated relatively free semigroup in the variety generated by all combinatorial strictly 0-simple semigroups.

ON FREE ALGEBRAS IN VARIETIES GENERATED BY ITERATED SEMIDIRECT PRODUCTS OF SEMILATTICES

We present a new solution of the word problem of free algebras in varieties generated by iterated semidirect products of semilattices. As a consequence, we provide asymptotical bounds for free spectra of these varieties. In particular, each finite -trivial (and, dually, each finite -trivial) semigroup has a free spectrum whose logarithm is bounded above by a polynomial function.

An assertion concerning functionally complete algebras and NP-completeness

In a paper published in J. ACM in 1990, Tobias Nipkow asserted that the problem of deciding whether or not an equation over a nontrivial functionally complete algebra has a solution is NP-complete. However, close examination of the reduction used shows that only a weaker theorem follows from his proof, namely that deciding whether or not a system of equations has a solution is NP-complete over such an algebra. Nevertheless, the statement of Nipkow is true as shown here. As a corollary of the proof we obtain that it is coNP-complete to decide whether or not an equation is an identity over a nontrivial functionally complete algebra.

On Rings with Few Orbits

In this article we investigate the structure of rings with some strong symmetry condition.