Automorphism groups of some variants of lattices

In this paper we consider variants of the power set and the lattice of subspaces and study automorphism groups of these variants. We obtain irreducible generating sets for variants of subsets of a finite set lattice and subspaces of a finite vector space lattice. We prove that automorphism group of the variant of subsets of a finite set lattice is a wreath product of two symmetric permutation groups such as first of this groups acts on subsets. The automorphism group of the variant of the subspace of a finite vector space lattice is a natural generalization of the wreath product. The first multiplier of this generalized wreath product is the automorphism group of subspaces lattice and the second is defined by the certain set of symmetric groups.

Permutations of zero-sumsets in a finite vector space

In this paper, we consider a finite-dimensional vector space {{\mathcal{P}}} over the Galois field {\operatorname{GF}(p)}, with p being an odd prime, and the family {{\mathcal{B}}_{k}^{x}} of all k-sets of elements of {\mathcal{P}} summing up to a given element x. The main result of the paper is the characterization, for {x=0}, of the permutations of {\mathcal{P}} inducing permutations of {{\mathcal{B}}_{k}^{0}} as the invertible linear mappings of the vector space {\mathcal{P}} if p does not divide k, and as the invertible affinities of the affine space {\mathcal{P}} if p divides k. The same question is answered also in the case where the elements of the k-sets are required to be all nonzero, and, in fact, the two cases prove to be intrinsically inseparable.

On graphs with same metric and upper dimension

The metric representation of a vertex [Formula: see text] of a graph [Formula: see text] is a finite vector representing distances of [Formula: see text] with respect to vertices of some ordered subset [Formula: see text]. The set [Formula: see text] is called a minimal resolving set if no proper subset of [Formula: see text] gives distinct representations for all vertices of [Formula: see text]. The metric dimension of [Formula: see text] is the cardinality of the smallest (with respect to its cardinality) minimal resolving set and upper dimension is the cardinality of the largest minimal resolving set. We show the existence of graphs for which metric dimension equals upper dimension. We found an error in a result, defining the metric dimension of join of path and totally disconnected graph, of the paper by Shahida and Sunitha [On the metric dimension of join of a graph with empty graph ([Formula: see text]), Electron. Notes Discrete Math. 63 (2017) 435–445] and we give the correct form of the theorem and its proof.

Covering codes of a graph associated to a finite vector space

UDC 512.5 In this paper, we investigate the problem of covering the vertices of a graph associated to a finite vector space as introduced by Das [Commun. Algebra, <strong>44</strong>, 3918 – 3926 (2016)], such that we can uniquely identify any vertex by examining the vertices that cover it. We use locating-dominating sets and identifying codes, which are closely related concepts for this purpose. We find the location-domination number and the identifying number of the graph and study the exchange property for locating-dominating sets and identifying codes.

Non-representable relation algebras from vector spaces

Extending a construction of Andreka, Givant, and Nemeti (2019), we construct some finite vector spaces and use them to build finite non-representable relation algebras. They are simple, measurable, and persistently finite, and they validate arbitrary finite sets of equations that are valid in the variety RRA of representable relation algebras. It follows that there is no finitely axiomatisable class of relation algebras that contains RRA and validates every equation that is both valid in RRA and preserved by completions of relation algebras. Consequently, the variety generated by the completions of representable relation algebras is not finitely axiomatisable. This answers a question of Maddux (2018).