ORDER COMPATIBILITY FOR CAUCHY SPACES AND CONVERGENCE SPACES

A Cauchy structure and a preorder on the same set are said to be compatible 
if both arise from the same quasi-uniform convergence structure on X. Howover, there are two natural ways to derive the former structures from the latter, leading 
to strong and weak notions of order compatibility for Cauchy spaces. These in turn lead to characterizations of strong and weak order compatibility for convergence 
spaces.

preordered Cauchy spaces of Section I.The second leads to a simple characterization of weakly preordered convergence spaces.l'ne terminology and noaio of [2] will be used ill his paper inhou further reference.However we shall always assume in this present paper that is a preorder (i.e., a transitive, reflexive relation).
From Definitions 2.1 and 2.2 of [2], we obtain the definitions of preordered uniform convergence space and preordered Cauchy space by simply ussuming that is a preorder instead of a partial order.The latter term will be abbreviated "p.c.s".
A p.c.s. (X, , ) for which is a partial order is called an ordered Cauchy pace (abbreviated "o.c.s.")In a p.c.s. (X, , ), we have "strong" compatibility between @ and because of the requirement in Definition 2.1, [2], that the filter <> on X x X generated by be in o.The relaxation of this requirement leads to the "weak" compatibility studied in the next section.
In this section, we consider triples of the form (X, @ ,C ), where (X, is a preordered set and C a Cauchy structure on X.In determining when (X, ,C is a p.c.s., the preorder < on the set C of Cauchy filters, as defined on p. 486, [2], plays a vital role since it is required in formulating two of the following conditions (see also p. 487, [2]): (OC)I C whenever (OC) 2 If , C < , and < , then C; x < implies x _< y.
(0c) 3 The first condition defines "local convexity" for a Cauchy structure on a preordered set; recall that ^() D -I) is the convex hull of The second condition asserts that the preorder < on is antisymmetric relative to Cauchy equivalence classes, and the third turns out to be equivalent to the order being closed in XXX.
It was shown in Theorem 2.9, [2], that when is a partial order, (X, , ) is an o.c.s, iff conditions (OC) I, (0C2), and (OC) 3 are all satisfied.In this section we show that when is a preorder, (X, , C) is a p.c.s, iff the same conditions hold.This task is made easier by the fact that all lemmas and Propositions in Section 2, [2] prior to Theorem 2.9 remain valid under the assumption that is a preorder rather than a partial or_der.Whereas the statement of Theorem 2.9, [2] remains valid when is a preorder, the proof of this theorem must be altered, since it makes explicit use of the assumption that is antisymmetric.Since, in particular Proposition 2.8, [2] is valid when is a preorder, the principal theorem of this section reduces to showing that (OC) I, (0C)2, and (OC)3 imply that (X, , is a p.c.s.Thus for the remainder of this section we assume that (X C is a preordered set with Cauchy structure which satisfies (OC)I' (OC)2' and (OC) 3.
Given (Y.)= an mquvalne rlan the elements of X is defined as follows: x y iff (x,y) ( -I.Let E {y ( X: x y} be the equivalence x class containing x, and let <E > be the filter of oversets of E LE4A I.I.For each x ( X, <Ex> y in (X,q C) for all y ( Ex.In particular, <E > C. x PROOF.It is obvious that E (y) N -l(y), for all y { X.Since E E Y x y and y, the conclusion follows by (OC) I.
LEMMA 1.2.If is a filter on X such that <> x then there is x X such that Ex ( " Furthermore, (C and y in (X, qc for all y Ex.PROOF.Choose F such that F x F N -I.clearly x ( F implies F c E The second statement follows by Lena 1. From Lemma 2 and (OC)3, the next lemma follows easily.
LEMMA 1.3.Assume that , are filters on X such that: <> c <> c , E and E ( .The following statements are equivalent: are PROOF.As we have noted previously, it is enough to prove that the three conditions are sufficient.From Proposition 2.6, (generalized to preordered sets), we see that (X, ,C) is a p.c.s, iff U(: ,C and ,C is compatible with C, where ,C A C is characterized in Proposition 2.4, [2]   and ,C (,C) v (,C)-I.In view of Proposition 2.7, [2], it is sufficient to show that ,C is compatible with C In other words, we must show, as in the proof of Theorem 2.9, [2] that x o,C implies C By Proposition 2.4, [2], we may assume x , where n -i @ j) x j)) <@>, and 5 < '' j n.If E for some j=l x x ( X, then C by Lemma i. Suppose E for all x X. 2hen there must x be an ultrafilter finer than such that < > x K. Otherwise, there must be (by Lemma ) ultrafilters K1 and K2, both finer than such that <> _c K1 x hl < > _c 2 x 2' and, for nonequivalent elements x and y, E x A and E K Since I x 2 D and K2 x I D 6, one can show that y 2 K1 K2 and K2 KI.But then (OC) 2 requires that K 1N K 2 C, and it follows from Lemma 3 that E K a contradiction.
x 2' Let be the set of all ultrafilters finer than such that < > x K The argument given in paragraphs 2 and 3 of the proof of Theorem 2.9, [2] leads to the conclusion that there is ( C such that for all ( K Thus if I { : ( K }, we conclude that I and i ( C If I the proof is complete.Assume instead that there is an ultrafilter finer than such that Then < > c x and so E 6 for some X x X by Lemma 2. A repetition of a previous argument shows that every such ultra- filter must contain E for the same x X.If ]L is the set of all such As in an earlier paragraph of the proof, we can deduce that K < and < K, from which it follows by (OC) 2 that X N C Thus I N 2 C, and the proof is complete.
PROPOSITION 1.5.If (X, , C) is a p.c.s., then is closed in X X.
PROOF.It is shown in Proposition 2.7, [2], that (OC) 3 is equivalent to the statement U{Q : @, Since <> o@,C and o,C is closed under compositions @=U{Do <> o: o,C U{ <> ,C where the last equality follows from 4.1.5,p. 301, [I], the closure being taken relative to the product convergence structure on X X derived from the convergence structure on X compatible with ,C By Proposition 2.6, [I], the latter convergence structure is precisely qc Let (X,q) be a convergence space (in the sense of Fischer), let C q be the set of all q-convergent filters, and let q(x) be the set of all filters which q-converge to x.It is well known that the following statements are equivalent: (a) There is a Cauchy structure C on X such that q q c (b)

C
q is a Cauchy structure on X.
(c) For x, y ( X, q(x) and q(y) are either equal or disjoint.
If (X,) is a preordered set and q a convergence structure on X, we define the triple (X, @, q) to be a preordered convergence space if there is a Cauchy structure C on X such that (X, ,C is a p.c.s, and q qc PROPOSITION 1.6.Let (X, be a preordered set, q a convergence structure on X.Then (X, @ q) is a preordered convergence space iff (X, C q) is a p.c.s.
PROOF.Let (X, @ q) be a preordered convergence space, and let (X, C be a Note that q D C. From the fact that (X, @, C) satisfies p.c.s, such that q q c (OC) and (OC) 3, it is easy to deduce that (X, @, cq) has the same properties.
Furthermore the latter space also satisfies (OC) 2, since for complete spaces (OC) 2 follows immediately from (OC) 3.
The converse argument is trivial.
THEOREM 1.7.Let (X, ) be a preordered set, q a convergence structure on X.
Then (X, , q) is a preordered convergence space iff (X, , q) is locally convex and @ is closed.
PROOF.Using Proposition 1.5, it is a simple matter to verify that the two properties specified for (X, , q) are equivalent to the assertion that (X, @, q) Since (X, , q) is complete, Theorem 1.4 and satisfies (OC) and (OC) 3.
Proposition 1.6 imply the desired conclusion.

WEAKLY PREORDERED CAUCHY SPACES (WEAK COMPATIBILITY).
As before, we assume that X is a set, a preorder on X, and a Cauchy structure on X. (X, , C) is defined to be a weakly preordered Cauchy space if there is a quasi-uniform convergence structure on X such that: (I) the uniform -I convergence structure v j derived from has C as its set of Cauchy filters; (2) = U{ }.This definition differs from that of preordered Cauchy space only in the condition < > , which is required for the latter but not for our present definition.We abbreviate weakly preordered Cauchy space by "w.p.c.s.".
Given a preorder on X, let Y be the quasi-uniform convergence structure on X with base consisting of all finite filter intersections of the form 0{x i x Yi (xi' y') ' i n}.If, in addition, C is a Cauchy structure on X, let T,C ^w here the lattice meet i taken in the lattice of all quasi-uniform convergence structures on X.Finally, we define a preorder on filters in C as follows: Q iff fl or there is (x,y) such that fl C and 9 As we shal see, I,C and Qplay the same role in the theory of weak compatibility that o,C and respectively, play in that of strong compatibility.PROPOSITION 2.1.Let (X,) be a preordered set with Cauchy structure Then Y, has a base of sets consisting of all finite intersections of the form { j xj j n}, where j j, j n.
n fl where PROOF Consider a composition of the fom I 2 " each i x i and there are two possibilities for i and i (I) "i i where (xi, yi) , or (2) i i' and i i C (i.e., 3i C )" By examining four possible cases, it is easy to verify that the existence of the composition i i+l implies that i Q i+l" Thus if I and n' we obtain for the entire chain of compositions Q and 61 %0...0 %.
We thus see that filters of the form indicated in the proposlton are in YC and a base for Y,C involves taking finite compositions and intersections of such filters.It can be shown by a straightforward set theoretic argument that any finite composition of filters of the indicated form can again be written as a finite intersection of filters of the same form, which is the desired conclusion PROPOSITION 2.2.A triple (X, , C) is a w.p.c.s, iff U{ : T, }, -I and C is compatible with , T,C v (T,) PROOF.Given the two conditions, (X, , C) is a w.p.c.s, according to the definition of this term.Conversely, assume that (X, , C) is a w.p.c.s, derived from a quasi-uniform convergence structure , and let v s-I be the associated uniform convergence structure Then it is easy to see that C , and since and C are compatible with C so is C Also D{0 : 6 o} u{ne : ^u} a u{ne:e ,e since (x,y) x x @ T, We next introduce conditions on a triple (X, ,C which lead to a character- ization of a w.p.c.s, similar to that given for a p.c.s, in Theorem 1.4.It turns out that only two such conditions are needed (woc)2 and (woc) 3 which are analogous to (OC) 2 and (OC)3, respectively.There is no form of "local convexity" involved in the characterization of weak compatibility, and so we shall later use (woc) to describe a single condition (not related to (OC)p which can replace both (woc) 2 and (wo) 3 (woc)2 If W , are in C, @, and Q then WA C.
(woc) 3 @ implies (x,y) E @.PROPOSITION 2. -1 (2) =P' where P , Te, V (Te,C (I).If (x,y) @, then x T, which implies (x,y) U{ : T, c }.If (x,y) G for some f ,e then x x ZG implying x T,c By Proposition 2.1, Q , and so (x,y) by (woe)3" (2) It is sufficient to show x E T implies , By ,e Proposition 2.1, we may assume x _DG= (.j x j)) , where j Q j, for j l,...,n.Let N, be distinct ultrafilters finer than Then there are indices i, j such that "i x,i c. and j x.j _ x , where i@"i and j@ j.This implies N @ and @N so that N by (woc)2.Thu Rearranging the indices if necessary, let l,...,m be the indices such that, if j _< m, then there are ultrafilters , finer than H such that x D__ j j, m where j @ '3 in g Let j A and 7 jlj.The reasoning of the preceding paragraph leads to the conclusion that 1 and that N for every ultrafilter _m H Thus which implies { C and the proof is complete.
We now introduce the condition (woc for a triple (X, , ).
It is significant that this condition is formulated entirely in terms of and q the convergence structure derived from THEOREM 2.5.(X, is a w.p.c.s, iff condition (woc) is satisfied.PROOF.In both directions of this proof, we use the characterization of a w.p.c.s, given in Theorem 2.4.