Coherence implies congruence-regularity (a local version)
The article considers a problem from Trokhimenko paper [13] concerning the study of abstract properties of commutations of operations and their connection with the Menger and Mann superpositions. Namely, abstract characterizations of some classes of operation algebras, whose signature consists of arbitrary families of commutations of operations, Menger and Mann superpositions and their various connections are found. Some unsolved problems are given at the end of the article.
We study commutative directoids with a greatest element, which can be equipped with antitone bijections in every principal filter. These can be axiomatized as algebras with two binary operations satisfying four identities. A minimal subvariety of this variety is described.
Commutative semigroups satisfying the equation and having only two -invariant congruences for an automorphism group are considered. Some classes of these semigroups are characterized and some other examples are constructed.
We prove that the lattice of varieties contains almost no compact elements.