REPRESENTATION OF DISTRIBUTIVE LATTICES BY MEANS OF ORDERED STONE SPACES

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1 REPRESENTATION OF DISTRIBUTIVE LATTICES BY MEANS OF ORDERED STONE SPACES H. A. PRIESTLEY 1. Introduction Stone, in [8], developed for distributive lattices a representation theory generalizing that for Boolean algebras. This he achieved by topologizing the set X of prime ideals of a distributive lattice A (with a zero element) by taking as a base {P a : aea} (where P a denotes the set of prime ideals of A not containing a), and by showing that the map a i-> P a is an isomorphism representing A as the lattice of all open compact subsets of its dual space X. The topological spaces which arise as duals of Boolean algebras may be characterized as those which are compact and totally disconnected (i.e. the Stone spaces); the corresponding purely topological characterization of the duals of distributive lattices obtained by Stone is less satisfactory. In the present paper we show that a much simpler characterization in terms of ordered topological spaces is possible. The representation theorem itself, and much of the duality theory consequent on it [8, 6], becomes more natural in this new setting, and certain results not previously known can be obtained. It is hoped to give in a later paper a more detailed exposition of those aspects of the theory barely mentioned here. I should like to thank my supervisor, Dr. D. A. Edwards, for some helpful suggestions and also Dr. M. J. Canfell for permission to quote from his unpublished thesis. 2. Preliminary definitions The notation concerning lattices is in accordance with [2], where all the necessary definitions and results may be found. The relevant theory of ordered spaces is less accessible, and will be summarized. We recall that a subset of a partially ordered set X is increasing if xse,y ^ x imply y e E; decreasing sets are defined analogously. If (X, &~, < ) is an ordered space (by which we shall mean a set X, with a topology &~, endowed with a partial order relation ^), then the set % consisting of the increasing sets in ST and the set S consisting of the decreasing sets in ^" each define a topology on X. (X, 3T, <;) is said to be totally order disconnected [Canfell (3)] if, given x,yex with x ^ y, then there exist disjoint ^"-clopen sets Ue<%, LzS such that yeu, XGL. Total order disconnectedness of (X, ST, ^) implies that 2T is T 2, and reduces to total disconnectedness when ^ is the trivial order: x ^ y if and only if x = y. 3. The representation theorem We shall consider only distributive lattices which have zero and identity elements. Lattice homomorphisms are consequently defined to be maps preserving these Received 18 December, [BULL. LONDON MATH. SOC, 2 (1970), ]

2 REPRESENTATION OF DISTRIBUTIVE LATTICES BY MEANS OF ORDERED STONE SPACES 187 elements in addition to preserving joins and meets. These restrictions are not essential; they are imposed here to prevent the theory's becoming obscured by minor technicalities. To obtain the representation of a distributive lattice A we shall use homomorphisms from A into a two element distributive lattice (with elements denoted by 0, 1) in preference to Stone's construction using prime ideals. The two approaches are equivalent [8; Theorem 5], but the topology we require is more easily defined in the context of homomorphisms. Hence let X be the set of all lattice homomorphisms /: A -> {0,1}; X is closed in the topology 2T induced by the product topology of {0, \} A and hence compact. We order X by defining, for /, g e X, f > g if and only iff (a) ^ g(a) for all aea. LEMMA. (X, 2T, ^) is totally order disconnected. Proof. Let/, gex,f g. Then there exists ae A such that g{a) = 1, f(a) = 0. The set U = {hex : h(a) = 1} is ^"-clopen, increasing and contains g, and X U is ^"-clopen, decreasing and contains/. We define the dual space of A to be the compact totally order disconnected space (X, y } <). Conversely, the dual lattice of a compact totally order disconnected space is by definition the set of all clopen increasing subsets with the obvious lattice operations, zero and identity. Our reformulation of Stone's representation theorem is the following. THEOREM 1. Let A be a distributive lattice with zero and identity and let (X, ST, < ) be its dual space. Then A is isomorphic to the dual lattice of (X, 2T, <=). Proof. Let A! be the dual lattice of (X, 9~, <) and define F by F(a) = {fex:f(a)=l}. For each aea, F{d) is a clopen increasing subset of X, and a standard computation shows that F : A -* A' is a homomorphism. To show F 1 : 1 we invoke the connection between homomorphisms and prime ideals, and a result due to Birkhoff stating that, for any two distinct elements a, b of A, there exists a prime ideal containing one of these elements but not the other [8; Theorem 6]. It remains to prove that F is onto A'. We define 0t = {F(a) : aea}, </> = {X-F(a) : aea}, and note that these families are closed under finite unions and intersections. Let U be clopen increasing in X; we require to show that Ue!%. If feu, gex U, then g >/. It follows that there exist disjoint sets R(f,g)eM, S{f,g)e9 > with fer(fg), ges(f,g). The compactness of X U enables this set to be expressed as a finite union of sets S(f,gi), g i ex U,i=l,...,m. Letting = U we have U = (J R(f).

3 188 H. A. PRIESTLEY Compactness of U allows this to be reduced to a finite union of elements of 0t, that is an element of 0t. This completes the proof. Theorem 1 has as a corollary the well-known representation of a finite distributive lattice as the set of all increasing subsets of a finite partially ordered set. By ignoring the order relation on the dual space (X, &~, <) we obtain a compact totally disconnected space (X, 2T}. It is easily proved from properties of ordered spaces that every clopen subset of (X, &~) is of the form U (Pt"Qd, where P h X Q { are clopen increasing subsets of (X,&~, <). Consequently the Boolean algebra B of clopen subsets of (X, 3") provides the minimal Boolean extension of A (unique up to isomorphism) (cf. [6; Theorem 2.1]). A equals its minimal Boolean extension if and only if every clopen subset of its dual space is increasing, or equivalently if and only if the order relation is trivial. Hence a necessary and sufficient condition for a distributive lattice to be a Boolean algebra is that its dual space have a trivial order (this may also be seen directly from the definition of the order). Other classes of distributive lattices can be characterized in terms of their dual spaces, for example chains (those distributive lattices whose dual spaces are totally ordered), relatively complemented lattices and certain closure algebras. 4. Duality theory Dual to Theorem 1 we have the following result characterizing those spaces which arise as duals of distributive lattices. THEOREM 2. Let (X, ZT, ^) be a compact totally order disconnected space and A its dual lattice and denote by (Y, ft\ ^') the dual space of A. Then (X, F, ^), (Y, &~', ^') are homeomorphic as topological spaces and isomorphic as partially ordered sets. Henceforth, if A is a distributive lattice with dual space X, A will be identified with the clopen increasing subsets of X. Associated with the duality theorems for Boolean algebras and Stone spaces is a variety of results expressing duality between homomorphisms and continuous maps, between ideals and open sets, etc. [4, 7]. For distributive lattices many of the corresponding results are known (Stone [8]; Nerode [6]), but become simpler when rephased in terms of ordered spaces. For part of Theorem 3, introduction, as in [6], of the minimal Boolean extensions provides a possible method of proof; however, a direct approach is probably simpler. We recall that a map 0 : X-> Y, where X, Y are partially ordered sets, is increasing if x t ^ x 2 in X implies <K*i) < <K*2) m Y- THEOREM 3. Let A lt A 2 be distributive lattices, X u X 2 their dual spaces. Then there exists a one-one correspondence between lattice homomorphisms f: A t -* A 2

4 REPRESENTATION OF DISTRIBUTIVE LATTICES BY MEANS OF ORDERED STONE SPACES 189 and continuous increasing maps (j): X 2 -> X l given by (4>(y))(fl) = y(f(a)) for all aea u yex 2. f is onto if and only if (j) is an order isomorphism, f is one-one if and only if (j) is onto. The duality of the categories C x (distributive lattices with 0, 1 and lattice homomorphisms) and C 2 (compact totally order disconnected spaces and continuous increasing maps), implied by Theorems 1, 2 and 3, provides the link between the theorem of Balbes [1] that the injective objects in C x are precisely the complete Boolean algebras and the theorem that the projective objects in C 2 are those elements of C 2 which have trivial order relation and are extremally disconnected. A one-one correspondence between ideals of the distributive lattice A and open increasing sets in its dual X is established by associating to each ideal the union of its members and to each open increasing subset the set of its clopen increasing subsets. To characterize the duals of prime ideals we adopt an approach (different from that used by Stone) which restores the symmetry lost in the transition from Boolean algebras to distributive lattices. Let B be the minimal Boolean extension of A, define A* to be the distributive lattice consisting of the complements in B of the elements of A, and consider the subset A x A* of B x B. By analogy with the concept of a bi-ideal in a pair of semi-algebras introduced by Canfell in [3], we define a bi-ideal in A x A* to be a set of the form/ x J with /, J ideals in A, A* respectively such that, for all i el, j e J, i v j # 1. A study of the properties of bi-ideals reveals that / x J is maximal (with respect to pairwise set inclusion) if and only if /, J are prime in A, A* respectively and J = {a' EA* : a$i}. Returning to the dual space of A, it is easily seen that the dual of a bi-ideal is a pair (U, L) e U x & such that U u L ^ X. Pairs of sets maximal with respect to this property are of the form U = X-d{x), L = X-i(x), where xel and d(x) = {yex: y^x}, i(x) = {yex : y > x}. Hence the prime ideals of A correspond to the open increasing sets X d(x),for xex. From this argument it also follows that the maximal bi-ideals are precisely those bi-ideals of the form (K n A) x (K n A*), where K is a maximal ideal in B. A further consequence is a theorem of Nachbin [5] to the effect that a distributive lattice is a Boolean algebra if and only if every prime ideal is maximal. From the duality for structure preserving maps follows a type of duality for the structures themselves. For example, A 2 is the quotient of A^ by an ideal if and only if X 2 is, in an appropriate sense, a subspace of X t. This result is related to Corollary of [6]. The dual problem (apparently not considered in the earlier literature) of finding the topological relation between X x and X 2 necessary and sufficient for A x to be a sublattice of A 2 has only a partial solution. To supply details would necessitate giving a discussion of quotients of ordered spaces which space does not permit.

5 190 REPRESENTATION OF DISTRIBUTIVE LATTICES BY MEANS OF ORDERED STONE SPACES References 1. R. Balbes, " Projective and injective distributive lattices ", Pacific J. Math., 21 (1967), G. Birkhoff, Lattice theory, American Mathematical Society Colloquium Publications XXV (3rd edition) (Providence, R.I., 1967). 3. M. J. Canfell, Semi-algebras and rings of continuous functions, Thesis, University of Edinburgh, P. Halmos, Boolean algebras, Van Nostrand Mathematical Studies 1 (Princeton, N.J., 1963). 5. L. Nachbin, " Une propriety caract&istique des algebres booleiennes ", Portugal Math., 6 (1947), A. Nerode, " Some Stone spaces and recursion theory ", Duke Math. J., 26 (1959), R. Sikorski, Boolean algebras (3rd edition) (Springer-Verlag, Berlin; Heidelberg, New York, 1969). 8. M. H. Stone, " Topological representations of distributive lattices and Brouwerian logics ", CasopisPSst. Math., 67 (1937), Mathematical Institute, Oxford.