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.
We develop a general axiomatic theory of algebraic pairs, which simultaneously generalizes several algebraic structures, in order to bypass negation as much as feasible. We investigate several classical theorems and notions in this setting including fractions, integral extensions, and Hilbert's Nullstellensatz. Finally, we study a notion of growth in this context.
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...