http://dspace.nbuv.gov.ua:80/handle/123456789/150382 2026-04-09T12:55:21Z 2026-04-09T12:55:21Z Abdioglu, C. Şahinkaya, S. KÖR, A. http://dspace.nbuv.gov.ua:80/handle/123456789/152357 2019-06-10T22:25:21Z 2014-01-01T00:00:00Z

Rigid, quasi-rigid and matrix rings with (σ,0)-multiplication Abdioglu, C.; Şahinkaya, S.; KÖR, A. Let R be a ring with an endomorphism σ. We introduce (σ, 0)-multiplication which is a generalization of the simple 0- multiplication. It is proved that for arbitrary positive integers m ≤ n and n ≥ 2, R[x; σ] is a reduced ring if and only if Sn,m(R) is a ring with (σ, 0)-multiplication.

2014-01-01T00:00:00Z Yegnanarayanan, V. http://dspace.nbuv.gov.ua:80/handle/123456789/152354 2019-06-10T22:25:18Z 2014-01-01T00:00:00Z

Chromatic number of graphs with special distance sets, I Yegnanarayanan, V. Given a subset D of positive integers, an integer distance graph is a graph G(Z, D) with the set Z of integers as vertex set and with an edge joining two vertices u and v if and only if |u−v| ∈ D. In this paper we consider the problem of determining the chromatic number of certain integer distance graphs G(Z, D)whose distance set D is either 1) a set of (n + 1) positive integers for which the nth power of the last is the sum of the nth powers of the previous terms, or 2) a set of pythagorean quadruples, or 3) a set of pythagorean n-tuples, or 4) a set of square distances, or 5) a set of abundant numbers or deficient numbers or carmichael numbers, or 6) a set of polytopic numbers, or 7) a set of happy numbers or lucky numbers, or 8) a set of Lucas numbers, or 9) a set of Ulam numbers, or 10) a set of weird numbers. Besides finding the chromatic number of a few specific distance graphs we also give useful upper and lower bounds for general cases. Further, we raise some open problems.

2014-01-01T00:00:00Z Umar, A. http://dspace.nbuv.gov.ua:80/handle/123456789/152350 2019-06-10T22:25:17Z 2014-01-01T00:00:00Z

Some combinatorial problems in the theory of partial transformation semigroups Umar, A. Let Xn = {1,2,…,n}. On a partial transformation α : Dom α ⊆ Xn → Im α ⊆ Xn of Xn the following parameters are defined: the breadth or width of α is ∣ Dom α ∣, the collapse of α is c(α) = ∣ ∪t∈Imα{tα⁻¹ :∣ tα⁻¹ ∣≥ 2} ∣, fix of α is f(α) = ∣ {x ∈ Xn : xα = x} ∣, the height of α is ∣ Imα ∣, and the right [left] waist of α is max(Imα) [min(Imα)]. The cardinalities of some equivalences defined by equalities of these parameters on Tn, the semigroup of full transformations of Xn, and Pn the semigroup of partial transformations of Xn and some of their notable subsemigroups that have been computed are gathered together and the open problems highlighted.

2014-01-01T00:00:00Z Protasov, I. Slobodianiuk, S. http://dspace.nbuv.gov.ua:80/handle/123456789/152346 2019-06-10T22:25:09Z 2014-01-01T00:00:00Z

On the subset combinatorics of G-spaces Protasov, I.; Slobodianiuk, S. Let G be a group and let X be a transitive G-space. We classify the subsets of X with respect to a translation invariant ideal J in the Boolean algebra of all subsets of X, introduce and apply the relative combinatorical derivations of subsets of X. Using the standard action of G on the Stone-ˇCech compactification βX of the discrete space X, we characterize the points p ∈ βX isolated in Gp and describe a size of a subset of X in terms of its ultracompanions in βX. We introduce and characterize scattered and sparse subsets of X from different points of view.

2014-01-01T00:00:00Z