can be strongly embedded into category of semigroups
In this paper we introduce the concept of an -representation of an algebra which is a common generalization of subdirect, full subdirect and weak direct representation of . Here we characterize such representations in terms of congruence relations.
Let be a type of algebras. A valuation of terms of type is a function assigning to each term of type a value . For , an identity of type is said to be -normal (with respect to valuation ) if either or both and have value . Taking with respect to the usual depth valuation of terms gives the well-known property of normality of identities. A variety is called -normal (with respect to the valuation ) if all its identities are -normal. For any variety , there is a least...
The concept of a -closed subset was introduced in [1] for an algebraic structure of type and a set of open formulas of the first order language . The set of all -closed subsets of forms a complete lattice whose properties were investigated in [1] and [2]. An algebraic structure is called - hamiltonian, if every non-empty -closed subset of is a class (block) of some congruence on ; is called - regular, if for every two , whenever they have a congruence class in common....
For an algebraic structure or type and a set of open formulas of the first order language we introduce the concept of -closed subsets of . The set of all -closed subsets forms a complete lattice. Algebraic structures , of type are called -isomorphic if . Examples of such -closed subsets are e.g. subalgebras of an algebra, ideals of a ring, ideals of a lattice, convex subsets of an ordered or quasiordered set etc. We study -isomorphic algebraic structures in dependence on the...