On Pseudo-simple Universal Algebras

A universal algebra is simple if and only if it has more than one element and there are no proper homomorphisms defined upon it. In this note we study a related notion : an algebra is pseudo-simple if and only if it has more than one element and is isomorphic to all its proper (i.e., non-one-element) homomorphic images. The two concepts simplicity and pseudo-simplicity are examples of concepts of irreducibility in universal algebra. Several of these notions of irreducibility may be equivalently expressed in terms of the lattice of congruence relations of the given algebra. Thus an algebra is simple if and only if the corresponding lattice is a chain of length two ; subdirectly irreducible if the lattice has a smallest nonzero element; and weakly sub-directly irreducible if the non-identity congruences themselves form a lattice. It is natural to inquire about a similar equivalent expression for the notion of pseudo-simplicity.1 In this note we characterize those lattices which can be the lattice of all congruence relations of a pseudo-simple algebra. We show that each of these lattices can also be the lattice of congruence relations on a non-pseudo-simple algebra. Thus there is no formulation of the notion of pseudosimplicity solely in terms of the congruence lattice. Where not otherwise stated, we shall use the notation of Birkhoff's book [2].