Purity and Equational Compactness of Projection Algebras

Abstract

The notions of purity and equational compactness of universal algebras have been studied by Banaschewski and Nelson. Also, Banaschewski deals with these notions in the special case of G-sets for a group G. In this paper we study these and related concepts in the category PRO of projection algebras, that is in N∞-sets, for the monoid N∞ with the binary operation m.n=min {m,n}. We show that every monomorphism in PRO is pure and hence every equationally compact projection algebra is in fact injective. Then, we introduce the notions of s-purity and s-compactness by which we characterize the retractions and hence equationally compact projection algebras. And, among other results, we show that equationally compact, injective, and complete projection algebras are the same. Finally, we characterize (pure-)essential monomorphisms and construct the Equationally Compact Hulls, equivalently the Injective Hulls, of projection algebras. These results, among other things, generalize the main results of Guili, regarding completeness and s-injectivity in the category PROs of separated projection algebras.