Under a multigraph it is meant in this paper a general incidence structure with finitely many points and blocks such that there are at least two blocks through any point and also at least two points on any block. Using submultigraphs with saturated points there are defined generating point sets, point bases and point skeletons. The main result is that the complement to any basis (skeleton) is a skeleton (basis).
We shall show that there exist sofic groups which are not locally embeddable into finite Moufang loops. These groups serve as counterexamples to a problem and two conjectures formulated in the paper by M. Vodička, P. Zlatoš (2019).
We are investigating quasigroup functional equation classification up to parastrophic equivalence [Sokhatsky F.M.: On classification of functional equations on quasigroups, Ukrainian Math. J. 56 (2004), no. 4, 1259–1266 (in Ukrainian)]. If functional equations are parastrophically equivalent, then their functional variables can be renamed in such a way that the obtained equations are equivalent, i.e., their solution sets are equal. There exist five classes of generalized distributive-like quasigroup...
In this paper we consider some special classes of Diophantine equations connected with McFarland's and Ma's conjectures about difference sets in abelian groups and we obtain an extension of known results.
arious methods of constructing nested ternary and quaternary efficiency balanced and variance balanced designs are proposed by applying some repetitions of treatments in all possible pairs of treatments. In these designs sub-blocks and super-blocks may form different p-ary designs, where sub-blocks have higher efficiency as compared to super-blocks, i.e., any two elementary treatment contrasts in the sub-blocks can be measured with higher efficiency than any two elementary contrasts in the super-block...