Mariusz Grech

Direct product of automorphism groups of colored graphs

The concrete representation problem asks if a permutation group G on a set X is equal (permutation isomorphic) to the automorphism group of some colored graph with vertex set X. In this paper we consider how the representability of the direct product of permutation groups is connected with the representability of the factors.

Irreducible varieties of commutative semigroups

Abstract In this paper we describe the varieties of commutative semigroups that are meet- and join-irreducible in the lattice of the varieties of commutative semigroups. We apply the method of A. Kisielewicz [Trans. Amer. Math. Soc. 342 (1994) 275–305]. This leads to investigation of the covering relation in the lattices of remainders and the algebraic structure of the remainders, involving permutation groups acting on the sequences of positive integers. In particular, along the way, we prove a theorem about existence of unique minimal generators for remainders, and provide algorithms to determine all the covers and dual covers of a given variety of commutative semigroups.

Automorphisms of the lattice of equational theories of commutative semigroups

In this paper we complete the study of the first-order definability in the lattice of equational theories of commutative semigroups started by A. Kisielewicz in [Trans. Amer. Math. Soc. 356 (2004), 3483-3504]. We describe the group of automorphisms of this lattice and characterize first-order definable theories, thus solving the problems posed by R. McKenzie and A. Kisielewicz.

Graphical cyclic permutation groups

We establish conditions for a permutation group generated by a single permutation of a prime power order to be an automorphism group of a graph or an edge-colored graph. This corrects and generalizes the results of the two papers on cyclic permutation groups published in 1978 and 1981 by S. P. Mohanty, M. R. Sridharan, and S. K. Shukla.

Symmetry groups of boolean functions

Abstract We prove that every abelian permutation group, but known exceptions, is the symmetry group of a boolean function. This solves the problem posed in the book by Clote and Kranakis. In fact, our result is proved for a larger class of permutation groups, namely, for all subgroups of direct sums of regular permutation groups.

The structure and definability in the lattice of equational theories of strongly permutative semigroups

In this paper, we study the structure and the first-order definability in the lattice L(SP ) of equational theories of strongly permutative semigroups, that is, semigroups satisfying a permutation identity x1 · · ·xn = xσ(1) · · ·xσ(n) with σ(1) > 1 and σ(n) < n. We show that each equational theory of such semigroups is described by five objects: an order filter, an equivalence relation, and three integers. We fully describe the lattice L(SP ); inclusion, operations ∨ and ∧, and covering relation. Using this description, we prove, in particular, that each individual theory of strongly permutative semigroups is definable, up to duality.

The graphical complexity of direct products of permutation groups

In this article, we improve known results, and, with one exceptional case, prove that when k≥3, the direct product of the automorphism groups of graphs whose edges are colored using k colors, is itself the automorphism group of a graph whose edges are colored using k colors. We have handled the case k = 2 in an earlier article. We prove similar results for directed edge-colored graphs.

Cyclic Automorphism Groups of Graphs and Edge-Colored Graphs

Abstract In this paper we describe the automorphism groups of graphs and edge-colored graphs that are cyclic as permutation groups. In addition, we show that every such group is the automorphism group of a complete graph whose edges are colored with 3 colors, and we characterize those groups that are automorphism groups of simple graphs.

2-closed abelian permutation groups

Abstract In this paper we demonstrate that the result by Zelikovskij concerning Konigs problem for abelian permutation groups, reported in a recent survey, is false. We propose in this place two results on 2-closed abelian permutation groups which concern the same topic in a more general setting.

Černý conjecture for edge-colored digraphs with few junctions

Abstract In this paper we consider the Cerný conjecture in terminology of colored digraphs corresponding to finite automata. We define a class of colored digraphs having a relatively small number of junctions between paths determined by different colors, and prove that digraphs in this class satisfy the Cerný conjecture. We argue that this yields not only a new class of automata for which the Cerný conjecture is verified, but also that our approach may be viewed as a new more systematic way to attack the Cerný conjecture in its generality, giving an insight into the complexity of the problem.