Eszter K. Horváth

On tolerance lattices of algebras in congruence modular varieties

We prove that the tolerance lattice TolA of an algebra A from a congruence modular variety V is 0-1 modular and satisfies the general disjointness property. If V is congruence distributive, then the lattice Tol A is pseudocomplemented. If V admits a majority term, then Tol A is 0-modular.

THE NUMBER OF TRIANGULAR ISLANDS ON A TRIANGULAR GRID

The aim of the present paper is to carry on the research of Czédli in determining the maximum number of rectangular islands on a rectangular grid. We estimate the maximum of the number of triangular islands on a triangular grid.

Elementary proof techniques for the maximum number of islands

Islands are combinatorial objects that can be intuitively defined on a board consisting of a finite number of cells. It is a fundamental property that two islands are either containing or disjoint. Czedli determined the maximum number of rectangular islands. Pluhar solved the same problem for bricks, and Horvath, Nemeth and Pluhar for triangular islands. Here, we give a much shorter proof for these results, and also for new, analogous results on toroidal and some other boards.

Cut approach to islands in rectangular fuzzy relations

The paper investigates fuzzy relations on a finite domain in the cutworthy framework, dealing with a new property coming from the information theory. If the domain of a relation is considered to be a table, then a rectangular subset of the domain whose values under this relation are greater than the values of all neighboring fields is called an island. Consequently, the so called rectangular fuzzy relations are introduced; their cuts consist of rectangles as sub-relations of the corresponding characteristic functions. A characterization theorem for rectangular fuzzy relations is proved. We also prove that for every fuzzy relation on a finite domain, there is a rectangular fuzzy relation with the same islands, and an algorithm for a construction of such fuzzy relations is presented. In addition, using methods developed for fuzzy structures and their cuts, we prove that for every fuzzy relation there is a lattice and a lattice valued relation whose cuts are precisely the islands of this relation. A connection of the notion of an island with formal concept analysis is presented.

Invariance groups of finite functions and orbit equivalence of permutation groups

Abstract Which subgroups of the symmetric group Sn arise as invariance groups of n-variable functions defined on a k-element domain? It appears that the higher the difference n-k, the more difficult it is to answer this question. For k ≤ n, the answer is easy: all subgroups of Sn are invariance groups. We give a complete answer in the cases k = n-1 and k = n-2, and we also give a partial answer in the general case: we describe invariance groups when n is much larger than n-k. The proof utilizes Galois connections and the corresponding closure operators on Sn, which turn out to provide a generalization of orbit equivalence of permutation groups. We also present some computational results, which show that all primitive groups except for the alternating groups arise as invariance groups of functions defined on a three-element domain.

One-Dimensional Czedli-Type Islands.

Summary The notion of an island has surfaced in recent algebra and coding theory research. Discrete versions provide interesting combinatorial problems. This paper presents the one-dimensional case with finitely many heights, a topic convenient for student research.

Cardinality of height function’s range in case of maximally many rectangular islands — computed by cuts

We deal with rectangular m×n boards of square cells, using the cut technics of the height function. We investigate combinatorial properties of this function, and in particular we give lower and upper bounds for the number of essentially different cuts. This number turns out to be the cardinality of the height function’s range, in case the height function has maximally many rectangular islands.

Cut approach to invariance groups of lattice-valued functions

This paper deals with lattice-valued n-variable functions on a k-element domain, considered as a generalization of lattice-valued Boolean functions. We investigate invariance groups of these functions, i.e., the group of such permutations that leaves the considered function invariant. We show that the invariance groups of lattice-valued functions depend only on the cuts of the function. Furthermore, we construct such lattice-valued Boolean function (and its generalization), the cuts of which represent all representable invariance groups.