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.

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.

On Forbidden Poset Problems in the Linear Lattice

In this note, we determine the maximum size of a $\{\mathrm{V}_{k}, \Lambda_{l}\}$-free family in the lattice of vector subspaces of a finite vector space both in the non-induced case as well as the induced case, for a large range of parameters $k$ and $l$. These results generalize earlier work by Shahriari and Yu. We also prove a general LYM-type lemma for the linear lattice which resolves a conjecture of Shahriari and Yu.

On resolvability of a graph associated to a finite vector space

In this paper, the resolving parameters such as metric dimension and partition dimension for the nonzero component graph, associated to a finite vector space, are discussed. The exact values of these parameters are determined. It is derived that the notions of metric dimension and locating-domination number coincide in the graph. Independent sets, introduced by Boutin [Determining sets, resolving set, and the exchange property, Graphs Combin. 25 (2009) 789–806], are studied in the graph. It is shown that the exchange property holds in the graph for minimal resolving sets with some exceptions. Consequently, a minimal resolving set of the graph is a basis for a matroid with the set [Formula: see text] of nonzero vectors of the vector space as the ground set. The matroid intersection problem for two matroids with [Formula: see text] as the ground set is also solved.

The Goldman-Rota Identity and the Grassmann Scheme

We inductively construct an explicit (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme.The key step is a constructive, linear algebraic interpretation of the Goldman-Rota recurrence for the number of subspaces of a finite vector space. This interpretation shows that the up operator on subspaces has an explicitly given recursive structure. Using the interpretation above we inductively construct an explicit orthogonal symmetric Jordan basis with respect to the up operator and write down the singular values, i.e., the ratio of the lengths of the successive vectors in the Jordan chains. The collection of all vectors in this basis of a fixed rank $m$ forms a (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme of $m$-dimensional subspaces. We also pose a bijective proof problem on the spanning trees of the Grassmann graphs.