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• W1987311030 abstract "Let X be a compact Hausdorff space equipped with a closed partial ordering. Let I be a linear ordering that either does not have a maximal element or does not have a minimal element. We further assume that (X, I) has the Tietze extension property for order preserving continuous functions from X to I. Denote by M(X, I) the lattice of order preserving continuous functions from X to I. We generalize a theorem of Kaplanski [K], and show that as a lattice alone, M(X, I) characterizes X as an ordered space. Throughout this paper we keep the following notations. An ordered space is a pair (X, < ) where X is a compact Hausdorff space, and < is a partial ordering on X such that {(x, Y)I x < y} is closed in X X X. We abbreviate (X, ) and denote it by X alone. I denotes a linear ordering: I is always regarded as a topological space with its interval topology. I denotes the Dedekind completion of I, namely the elements of I are those of I, and in addition, all pairs (L, R) where.L U R = I, L n R = 0, if a C L and b < a then b C L, and neither does L have a maximum nor does R have a minimum. A function f from a partially ordered set (A, < ) to a partially ordered set (B, < ) is order preserving (OP), if for every a, b c A: if a < b, then f(a) < f(b). M(X, I) denotes the lattice of OP continuous functions from X to I. The lattice operations A, V are respectively the pointwise minimum and the pointwise maximum; andf < g means that for every x C X,f(x) < g(x). We say that (X, I) has the Tietze extension property (TEP), if for every closed subset F of X and every OP continuous function f: F -I there is f C M( X, I) which extends f. Nachbin [N, Theorem 6] has shown that if I = Reals, then (X, I) has the TEP for every ordered space X. THEOREM 1. For i = 1,2, let (X,, Ii) have the TEP, and assume either I, has no maximal element, or no minimal element. Then, if M(X,, II), M(X2, I2) are lattice isomorphic, then there exists an order preserving homeomorphism between X, and X2. Theorem 1 generalizes [K] where X is assumed to have the trivial partial ordering, namely, every element is comparable just with itself. Our method of proof is very similar to Kaplansky's. As in the case of the lattice of all continuous functions, there is a natural way to associate every prime ideal in Received by the editors June 8, 1981 and, in revised form, March 9, 1982. 1980 Mathematics Subject Classification. Primary 54F05, 54C35; Secondary 54C20. (C1982 American Mathematical Society 0002-9939/82/0000-U3 10/$02.75" @default.
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• W1987311030 title "Lattices of continuous monotonic functions" @default.
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• W1987311030 doi "https://doi.org/10.1090/s0002-9939-1982-0674106-5" @default.
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