Journal für die reine und angewandte Mathematik

Injective hulls in the category of distributive lattices

By B, Banaschewski and G, Bruns at Hamilton/Ont. To H, Rohrbach on his 6jth Birthday

Introduction

This paper deals primarily with the relationship between the notions of injectivity and essential extension in the category D of all distributive lattices and their lattice homomorphisms, the main point being that in D the situation regarding these notions is fully analogous to what holds in the category of all left modules over a ring [4] or, for that matter, in the category of all partially ordered sets and order preserving mappings [2] (Propositions 2 and 3).

An interesting aspect of the results presented here seems to us the manner in which Boolean lattices make their appearance. Thus, it is proved that every distributiл7^e lattice has an essential extension which is Boolean (Proposition 1), and the injective hulls in D are shown to be the MacNeille completions of such Boolean essential extensions ( position 3).

The paper concludes with a discussion of injective hulls and essential extension of some particular types of distributive lattices and a result, already announced without proof in [2], which shows that essential extensions behave very badly in the category of all lattices and lattice homomorphisms^).

1 . Preliminaries

In the following, D will be the category of all distributive lattices and their lattice homomorphisms. General categorical notions will be used as in [6] and lattice theoretic concepts as in [3]. The complete atomic Boolean lattices will be called power set lattices.

One readily sees that the monomorphisms of D are precisely the one-to-one morphisms, and that any monomorphism /: L-> Af in D is actually an embedding^ i- ©•? / decomposes into an isomorphism from L to the (set theoretically, not categorically defined) sublattice Im (/) of M followed by the natural homomorphism Im (/) -^ M given by the set theoretic natural injection. A monomorphism /: L-> M is called essential if any g: M -> N for which ^ • / is a monomorphism is itself a monomorphism.

^ ) The main results of this paper were presented at the meeting on universal algebra held at Oberwolfach, Germany, in July 1966. Financial support from the National Eesearch Council of Canada on this occasion is gratefully acknowledged.