A multiplication of e-varieties of regular -solid semigroups by inverse semigroup varieties is described both semantically and syntactically. The associativity of the multiplication is also proved.
0. Introduction. Besides being of intrinsic interest, cylindric (semi-) lattices arise naturally from the study of dependencies in relational databases; the polynomials on a cylindric semilattice are closely related to the queries obtainable from project-join mappings on a relational database (cf. [D] for references). This note is intended to initiate the study of these structures, and only a few, rather basic results will be given. Some problems at the end will hopefully stimulate further research....
The paper contains two remarks on finite sets of groupoid terms closed under subterms and the application of unifying pairs.
The idempotent modification of a group is always a subdirectly irreducible algebra.
A variety is called normal if no laws of the form are valid in it where is a variable and is not a variable. Let denote the lattice of all varieties of monounary algebras and let be a non-trivial non-normal element of . Then is of the form with some . It is shown that the smallest normal variety containing is contained in for every where denotes the operator of forming choice algebras. Moreover, it is proved that the sublattice of consisting of all normal elements of...
It is shown that in a finitely decidable equational class, the solvable radical of any finite subdirectly irreducible member is comparable to all congruences of the irreducible if the type of the monolith is 2. In the type 1 case we establish that the centralizer of the monolith is strongly solvable.
We prove a theorem (for arbitrary ring varieties and, in a stronger form, for varieties of associative rings) which basically reduces the problem of a description of varieties with distributive subvariety lattice to the case of algebras over a finite prime field.
A diagrammatic statement is developed for the generalized semidistributive law in case of single algebras assuming that their congruences are permutable. Without permutable congruences, a diagrammatic statement is developed for the ∧-semidistributive law.