Representation of Universal Algebras by Sheaves

It is proved that every (universal) algebra A with distributive and permutable structure lattice is isomorphic with the algebra of all global sections with compact supports of a sheaf of homomorphic images of A over a topological space. This completely generalises the corresponding result of Klaus Keimel for /-rings. Introduction. The following note is a preliminary report on the investigations undertaken by the author towards representing universal algebras by (continuous) sections of sheaves.. The object of the present paper is to show that KeimePs general representation theorem [3, 3-3] for /-rings is valid for any universal algebra, with distributive and permutable structure lattice (see Theorem 2 below). The results in the general case are under investigation and will be communicated later. In §1 we define the (irreducible) spectrum of any (universal) algebra with distributive structure lattice and state its properties, whose proofs are routine. In §2 we give a general construction of sheaves of algebras over a topological space, along the lines of Keimel [3] and apply this to get a general representation theorem for an algebra, with.a distributive and permutable structure lattice. The crucial point of this note is Lemma 3, which replaces Lemma 1.11 of Keimel [3] where extensive use is made of the lring structure. Throughout this paper, A denotes an fi-algebra (algebra of type 0, see [2, p. 33]) and x(A) denotes its structure lattice. We refer to [4] for elementary basic properties of sheaves and [l] for preliminaries on sheaves of fl-a Ige bras. 1. The spectrum of an il-algebra. Let £(A) be distributive; as usual, we call a proper congruence qo on A (meet) irreducible iff, for any 0,6 e £(A), 6: n 62 Ç f> implies either 6, Ç <h or $2 Ç <ß. The set X of all irreducible congruences on A becomes a topological space by taking ÍXJ 6 e Received by the editors May 30, 1973. AMS (MOS) subject classifications (1970). Primary 18F20. Copyright © 1974, American Mathematical Society 55 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use