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.