Injective and Projective Heyting Algebras
Raymond Balbes, Alfred Horn · Transactions of the American Mathematical Society · 1970
The determination of the injective and projective members of a category is usually a challenging problem and adds to knowledge of the category.In this paper we consider these questions for the category of Heyting algebras.There has been a lack of uniformity in terminology in recent years.In [6] Heyting algebras are referred to as pseudo-Boolean algebras, and in [1] they are called Brouwerian lattices.We would argue for retaining the name Heyting algebras for the reasons given in [2, p. 162], reserving the name Brouwerian algebras for the algebras dual to Heyting algebras.The fundamental paper on Brouwerian algebras (and therefore Heyting algebras) is [4].We shall show that a Heyting algebra is injective if and only if it is a complete Boolean algebra.The determination of projective Heyting algebras is, as usual, more difficult.We shall characterize all projective Heyting algebras which are finite or are chains.Some other results on projective algebras are given.1. Definitions and preliminaries.If S is a subset of a lattice L, then 2(5) denotes the least upper bound in L of S, and fl(S) denotes the greatest lower bound of S. If S={x, y}, then 2(S) and 11(5') are written x+y and xy.The largest and smallest elements of a lattice L are denoted by 1 (or 1L) and 0 respectively.A Heyting algebra is a lattice with 0 such that for any elements a, b there is an element a->b such that axúb if and only if x^a->¿>.We use -¡a to denote a -> 0. Every Heyting algebra is a distributive lattice.The lattice of all open sets of a topological space is a Heyting algebra.We state without proof some elementary properties of Heyting algebras.Theorem 1.1.If a, b, and c are any elements of a Heyting algebra, then the following hold:(1) a = b if and only ifa->è=l.(2) l->a = a.(3) a->b^b.(4) Ifb = c, then a^-b = a-^-c andc -^a = b -^-a.