Formations, bihomorphisms and natural transformations

Abstract

Given a variety ν and ν-algebras A and B, an algebraic formationF: A ⇉ B is a ν-homomorphism FL R × A → B, for some ν-algebra R, and the resulting functions F (r,-): A → B for r ∈ R are termed formable. Firstly, as motivation for the study of algebraic formations, categorical formations and their relationship with natural transformations are explained. Then, formations and formable functions are described for some common varieties of algebras, including semilattices, lattices, groups, and implication algebras. Some of their general properties are investigated for congruence modular varieties, including the description of a uniform congruence which provides information on the structure of B.