An Easy Way to Minimal Algebras

A finite algebra C is called minimal with respect to a pair δ<θ of its congruences if every unary polynomial f of C is either a permutation, or f(θ)⊆δ. It is the basic idea of tame congruence theory developed by Ralph McKenzie and David Hobby [7] to describe finite algebras via minimal algebras that sit inside them. As shown in [7] minimal algebras have a very restricted structure. This paper presents a new tool, the Twin Lemma, which makes it possible to give short proofs of some of these structure theorems. This part can be read as an alternative introduction to the theory. Our method yields new information in the type 1 case, and is especially useful in describing E-minimal algebras (that is, algebras that are minimal with respect to every prime congruence quotient). We complete their theory given in [7] by proving a structure theorem for the type 1 case. Finally we show that if an algebra is minimal with respect to two quotients, then the two types are the same, and if this type is 2, 3, or 4, then the bodies are also equal.