Formations, bihomomorphisms and natural transformations

Given a variety V and V-algebras A and B, an algebraic formation F: A paired right arrows B is a V-homomorphism F: R x A --> B, for some V-algebra R, and the resulting functions F (r, (-)) : A --> B for r is an element of 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.