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.

Critical algebras and the Frattini congruence, II

We prove that any finite subdirectly irreducible algbra in a congruence modular variety with trivial Frattini congruence is critical. We also show that if A and B are critical algebras which generate the same congruence modular variety, then the variety generated by the proper sections of A equals the variety generated by the proper sections of B.

Finite algebras that generate an injectively complete modular variety

We extend Kollár's result on finitely generated, injectively complete congruence distributive varieties to the congruence modular setting. By doing so we show that, given any finite algebra A of finite type, there is an algorithm to decide whether V(A) is an injectively complete, congruence modular variety.

Relatively congruence distributive subquasivarieties of a congruence modular variety

We characterise the relatively congruence distributive subquasivarieties of a modular variety using the modular commutator. Our characterisation allows us to extend the results of Dziobiak concerning relatively congruence distributive quasivarieties of nonassociative R-algebras.

Tolerances and Commutators on lattices

The commutator has the following order theoretic properties: [α, β] ≦ α ∧ β, [α, β] = [β α],[α1 ∨ α2,β] = [α1, β] ∨ [α2, β] for congruences α, β ∈ Con A of an algebra A in a congruence modular variety generalising the original concept in group theory. A tolerance of a lattice L is a reflexive and symmetric sublattice of L2. We show that to every commutator [ , ] of Con A corresponds a ∧-subsemilattice of the lattice of tolerances of Con A. It can be shown that A in a congruence modular variety is nilpotent if |con A| > 2 and Con A is simple.