Integral Representation and the Computation of Multiple Combinatorial Sums from Hall’s Commutator Theory

In this paper we prove a series of combinatorial identities arising from computing the exponents of the commutators in P. Hall’s collection formula. We also compute a sum in closed form that arises from using the collection formula in Chevalley groups for solving B. A. F. Wehrfritz problem on the regularity of their Sylow subgroups

Medial and semimedial left quasigroups

In this paper, we investigate the class of semimedial left quasigroups, a class that properly contains racks and medial left quasigroups. We extend most of the results about commutator theory for racks collected in [M. Bonatto and D. Stanovský, Commutator theory for racks and quandles, preprint (2019), arXiv:1902.08980 .] and we provide a structure theorem for finite medial connected racks.

The annihilation graphs of commutator posets and lattices with respect to an element

We propose a new, widely generalized context for the study of the zero-divisor type (annihilating-ideal) graphs, where the vertices of graphs are not elements/ideals of a commutative ring, but elements of an abstract ordered set [lattice] (imitating the lattice of ideals of a ring), equipped with a commutative (not necessarily associative) binary operation (imitating the product of ideals of a ring). We discuss, when [Formula: see text] (the annihilation graph of the commutator poset [lattice] [Formula: see text] with respect to an element [Formula: see text]) is a complete bipartite graph together with some of its other graph-theoretic properties. In contrast to the case of rings, we construct a commutator poset whose [Formula: see text] contains a cut-point. We provide some examples to show that some conditions are not superfluous assumptions. We also give some examples of a large class of lattices, such as the lattice of ideals of a commutative ring, the lattice of normal subgroups of a group, and the lattice of all congruences on an algebra in a variety (congruence modular variety) by using the commutators as the multiplicative binary operation on these lattices. This shows that how the commutator theory can define and unify many zero-divisor type graphs of different algebraic structures as a special case of this paper.