It is well known that every complete lattice can be considered as a complete lattice of closed sets with respect to appropriate closure operator. The theory of q-lattices as a natural generalization of lattices gives rise to a question whether a similar statement is true in the case of q-lattices. In the paper the so-called M-operators are introduced and it is shown that complete q-lattices are q-lattices of closed sets with respect to M-operators.
A vertex -coloring of a graph is a multiset -coloring if for every edge , where and denote the multisets of colors of the neighbors of and , respectively. The minimum for which has a multiset -coloring is the multiset chromatic number of . For an integer , the -corona of a graph , , is the graph obtained from by adding, for each vertex in , new neighbors which are end-vertices. In this paper, the multiset chromatic numbers are determined for -coronas of all complete...
A vertex coloring of a graph G is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum k for which G has a multiset k-coloring is the multiset chromatic number χₘ(G) of G. For every graph G, χₘ(G) is bounded above by its chromatic number χ(G). The multiset chromatic numbers of regular graphs are investigated. It is shown that for every pair k, r of integers with 2 ≤ k ≤ r - 1, there exists an r-regular graph with multiset chromatic...
A well known result of R. Dedekind states that a lattice is nonmodular if and only if it has a sublattice isomorphic to . Similarly a lattice is nondistributive if and only if it has a sublattice isomorphic to or (see [11]). Recently a few results in this spirit were obtained involving the number of polynomials of an algebra (see e.g. [1], [3], [5], [6]). In this paper we prove that a nondistributive Steiner quasigroup (G,·) has at least 21 essentially ternary polynomials (which improves the...
We improve known bounds for the maximum number of pairwise disjoint arithmetic progressions using distinct moduli less than x. We close the gap between upper and lower bounds even further under the assumption of a conjecture from combinatorics about Δ-systems (also known as sunflowers).