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On the cardinality of some lattices
Volume 3, number 3
INFORMATIONPROCESSINGLE’ITERS
Yanuary1975
ONTHECARDINALITYOFSOMELATTICES ChiharuHOSONO Department o,fMatherustics, Kyoto University, Kyoto, Japan Received 12 August 1974 mathematical theory of computation
Recently Scott has vigorously developed a mathematical theory of computation based on lattice theory. Scott has shown that thereis a complete lattice which is isomorphic to the lattice consisting of all of its continuous functions. Such a latticeis interesting because it is a model of the type-freekalculus. The readeris referredto refs. [1,2] for background material and additional details. Jn this paper we show that such a lattice must have morethan countably many elements. We begin with some deftitions. , Notation. J:or a complete lattice D we denote the orderingrelationby C , the least upperbound operz tion by U p the greatestlower bound operationby n , and the maximum(minimum)element of D by TD(l&;forxandyinD,xIymeansxC yand x #y. L&$?nitiun.A subset X of a complete lattice D is said to be directed if ad only if everyfinite subset of X has an upper bound in X. A subset U of D is said to be open if and only if it satisfiesthe following two conditions. (l)I~xEUanrlx~ y,thenyEU. (2) For any d&ted set X, i U X E U, then there exc istss0mexEXnJ.L A sequence x0, xl,. . . of D is called a chain when x, C x,,+~ holds for all n 0. A function f: D+ D is said to be monotonic if and only iffix) GZ fiy) when XTP_y. For any complete lattice D, [D + i’] denotes the complete lattice consisting of all continu ous functions (i;1the sense of the from D to D, where f!G g if and forallxED.
Remark. It is well known that f: D + D is continuous if and only iff(U X) = U f(x) for every directed set XC D. Lemma 1. Jf D is a complete lattice which has at least two elements and there exists a lattice isomorphism a: [D + D] + D, then there exists a chain x0, x1,. . . Xn+a for all n Z 0. WithX,1 hxf. Define q: D + [D + D] such that *(a) = fafor aEDwheref,(x)=awhenx=Io,and&(x)=TD otherwise. Then @ is obviously one to one monotonic, so Q 0 JI is also one to one monotonic. By assumption D has at least two elements, so 1 Jo+01 cs G(.io). Therefore @(lJ,,,J)=L, EE QO@(I,). Let Xn = (a 0 \t)“(l&, then eviderrtly x, c xn+l for n Z 0 because Qi0 q is one to one monotonic, and x0 II x1. Therefore the chain x0, xl,. . . is a required one. Lemma 2. Let X = {x,r] be a countable subset of D such that x, c x~+~, then any monotonic function fi X + X can be extended to a continuous function f-:b+D. Roof. Givenf: X + f: D + D such that
X which is monotonic, we define .
F(x)= XE LJI/ zExFU f(z) where U ranges ov:r the open subsets of D. The continuity ofj%an be easily seen by using the Remark. Next for x, E X, it is easy to show (x,) C f(x,), In order to show the cenverse, take an open set (zl I (z L_ x, _l)i when n + 0, take D when n = 0; for this open set U, we have
r
3, nwnber 3
INFOR nATION PROCESSINGLETTERS
e cardinality of the set ofall monotonic ram N to N is not less than that of 2N, the set of natural numbers. ;o), define j+ N -+N an:yw&et vowp(J) = max {zGEV(z G x) where we de&e then fv is monotonic and fiv # .frr when
m llf D is a complete iattice \F;hiichis isomor), and has at least two elements, then 0fD is not less than that of P. 1, WChave a chain x0, ~1, . X = {x0, xl, . e.), and define ) = n. Every function ia 3 has a correspondiig n ,K By Lemma 2, & gv0g” extended to an element of [D -+Id]. Since D phic to p +D]., we have the theorem. l
68
l
January 1975
Ck~&ry. D, (see ref. [2] ) is not countable. RFOO1f: Trivial. Gmment by the referee. As a consequence of this theorem, one cannot expect to use Scott’s construction (as it stands now> to find a model in which all elements are effective computable. Although Scott’s models are intended to represent kexpressions, noncomputable ele:ments creep in.
Referem [ 1f B. Scott, Outline of a Mathematical Theory of Computation, Techn. McinographPRG-2, Oxford Univ. Comp. Lab. (1970). [2 J II. Scott, Continuous Lattices, Techn. Monograph PRG-7, Oxford Univ. Comp. Lab. (1970). [3j D. Scott, Data type5 as Lattices, to appear (1974).
INFORMATIONPROCESSINGLE’ITERS
Yanuary1975
ONTHECARDINALITYOFSOMELATTICES ChiharuHOSONO Department o,fMatherustics, Kyoto University, Kyoto, Japan Received 12 August 1974 mathematical theory of computation
Recently Scott has vigorously developed a mathematical theory of computation based on lattice theory. Scott has shown that thereis a complete lattice which is isomorphic to the lattice consisting of all of its continuous functions. Such a latticeis interesting because it is a model of the type-freekalculus. The readeris referredto refs. [1,2] for background material and additional details. Jn this paper we show that such a lattice must have morethan countably many elements. We begin with some deftitions. , Notation. J:or a complete lattice D we denote the orderingrelationby C , the least upperbound operz tion by U p the greatestlower bound operationby n , and the maximum(minimum)element of D by TD(l&;forxandyinD,xIymeansxC yand x #y. L&$?nitiun.A subset X of a complete lattice D is said to be directed if ad only if everyfinite subset of X has an upper bound in X. A subset U of D is said to be open if and only if it satisfiesthe following two conditions. (l)I~xEUanrlx~ y,thenyEU. (2) For any d&ted set X, i U X E U, then there exc istss0mexEXnJ.L A sequence x0, xl,. . . of D is called a chain when x, C x,,+~ holds for all n 0. A function f: D+ D is said to be monotonic if and only iffix) GZ fiy) when XTP_y. For any complete lattice D, [D + i’] denotes the complete lattice consisting of all continu ous functions (i;1the sense of the from D to D, where f!G g if and forallxED.
Remark. It is well known that f: D + D is continuous if and only iff(U X) = U f(x) for every directed set XC D. Lemma 1. Jf D is a complete lattice which has at least two elements and there exists a lattice isomorphism a: [D + D] + D, then there exists a chain x0, x1,. . . Xn+a for all n Z 0. WithX,1 hxf. Define q: D + [D + D] such that *(a) = fafor aEDwheref,(x)=awhenx=Io,and&(x)=TD otherwise. Then @ is obviously one to one monotonic, so Q 0 JI is also one to one monotonic. By assumption D has at least two elements, so 1 Jo+01 cs G(.io). Therefore @(lJ,,,J)=L, EE QO@(I,). Let Xn = (a 0 \t)“(l&, then eviderrtly x, c xn+l for n Z 0 because Qi0 q is one to one monotonic, and x0 II x1. Therefore the chain x0, xl,. . . is a required one. Lemma 2. Let X = {x,r] be a countable subset of D such that x, c x~+~, then any monotonic function fi X + X can be extended to a continuous function f-:b+D. Roof. Givenf: X + f: D + D such that
X which is monotonic, we define .
F(x)= XE LJI/ zExFU f(z) where U ranges ov:r the open subsets of D. The continuity ofj%an be easily seen by using the Remark. Next for x, E X, it is easy to show (x,) C f(x,), In order to show the cenverse, take an open set (zl I (z L_ x, _l)i when n + 0, take D when n = 0; for this open set U, we have
r
3, nwnber 3
INFOR nATION PROCESSINGLETTERS
e cardinality of the set ofall monotonic ram N to N is not less than that of 2N, the set of natural numbers. ;o), define j+ N -+N an:yw&et vowp(J) = max {zGEV(z G x) where we de&e then fv is monotonic and fiv # .frr when
m llf D is a complete iattice \F;hiichis isomor), and has at least two elements, then 0fD is not less than that of P. 1, WChave a chain x0, ~1, . X = {x0, xl, . e.), and define ) = n. Every function ia 3 has a correspondiig n ,K By Lemma 2, & gv0g” extended to an element of [D -+Id]. Since D phic to p +D]., we have the theorem. l
68
l
January 1975
Ck~&ry. D, (see ref. [2] ) is not countable. RFOO1f: Trivial. Gmment by the referee. As a consequence of this theorem, one cannot expect to use Scott’s construction (as it stands now> to find a model in which all elements are effective computable. Although Scott’s models are intended to represent kexpressions, noncomputable ele:ments creep in.
Referem [ 1f B. Scott, Outline of a Mathematical Theory of Computation, Techn. McinographPRG-2, Oxford Univ. Comp. Lab. (1970). [2 J II. Scott, Continuous Lattices, Techn. Monograph PRG-7, Oxford Univ. Comp. Lab. (1970). [3j D. Scott, Data type5 as Lattices, to appear (1974).