Order-theoretical Characterizations of Countably Approximating Posets 1

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1 Int. J. Contemp. Math. Sciences, Vol. 9, 2014, no. 9, HIKARI Ltd, Order-theoretical Characterizations of Countably Approximating Posets 1 Xuxin Mao College of Science Nanjing Univ. of Aeronautics and Astronautics Nanjing , China Luoshan Xu 2 Department of Mathematics, Yangzhou University Yangzhou , China Copyright c 2014 Xuxin Mao and Luoshan Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, the concepts of principal ideal countably approximating posets and closed interval countably approximating posets are introduced. Some order-theoretical characterizations of countably approximating posets are given. It is proved that the principal ideal countably approximating posets are exactly the countably approximating posets. It is also proved that countably approximating posets with multiple joins for nonempty finite subsets are closed interval countably approximating posets and that closed interval countably approximating posets satisfying the controlling condition from below are countably approximating posets. Mathematics Subject Classifications: 06A11; 06B35 Keywords: countably directed set; countably approximating poset; principal ideal; closed interval 1 Supported by the NSF of China ( , ). 2 Corresponding author

2 448 Xuxin Mao and Luoshan Xu 1 Introduction The notion of continuous lattices as a model for the semantics of programming languages was introduced by Scott in [8]. Later, a more general notion of continuous posets was introduced and extensively studied (see [3]-[7]). The concepts of principal ideal continuity and closed interval continuity on posets were introduced by Xu and Lawson and proved to be equivalent to the common continuity on posets under certain conditions (see [5] and [10]). On the other hand, Lee in [6] introduced the concept of countably approximating lattices, a generalization of continuous lattices, and showed that this new larger class has many properties in common with continuous lattices. In [2], Han, Hong, Lee and Park further generalized the concept of countably approximating lattices to the concept of countably approximating posets and presented some topological characterizations of countably approximating posets. In this paper, making use of the ideas of [2] and [5], we introduce the concepts of principal ideal countably approximating posets and closed interval countably approximating posets. Some order-theoretical characterizations of countably approximating posets are presented Relations among countably approximating posets, principal ideal countably approximating posets and closed interval countably approximating posets are explored. 2 Preliminaries We quickly recall some basic notions and results (see, e.x., [1], [2] or [5]). Let (P, ) be a poset. A principal ideal (resp., principal filter) is a set of the form x = {y P y x} (resp., x = {y P x y}). A closed interval [x, y] is a set of the form x y for x y. Note that closed intervals are always nonempty. For X P, we write X = {y P x X, y x} and X = {y P x X, x y}. A subset X is a(n) lower set (resp., upper set) ifx = X (resp., X = X). We say that a is an upper bound of X if X a. The set of all upper bounds of X is denoted by ub(x). An element m ub(x) isaminimal upper bound of X if for all s ub(x) with s m, one has s = m. The set of all minimal upper bounds of X is denoted by mub(x). The supremum of X is the least upper bound of X and denoted by X or sup X. A subset D of P is directed if every finite subset of D has an upper bound in D. A subset D is countably directed if every countable subset of D has an upper bound in D. Clearly every countably directed set is directed but not vice versa. A poset P is a directed complete partially ordered set (dcpo, in short) if every directed subset of P has a supremum. A poset is said to have countably directed joins if every countably directed subset has a supremum. It is clear that if D is countably directed and itself is countable, then D has

3 Order-theoretical characterizations 449 a maximal element. By this observation, we see that every countable poset has countably directed joins and thus a poset having countably directed joins needn t be a dcpo. Definition 2.1. (see [1, 11]) Let P be a poset and x, y P. We say that x is way-below y or x approximates y, written x y if whenever D is a directed set that has a supremum sup D y, then there is some d D with x d. For each x P, we write x = {y P y x}. A poset is said to be continuous if every element is the directed supremum of elements that approximate it. A continuous poset which is also a complete lattice is called a continuous lattice. Definition 2.2. (see [2]) Let P be a poset and x, y P. We say that x is countably way-below y, written x c y if for any countably directed subset D of P with sup D y, there is some d D with x d. For each x P,we write c x = {y P y c x}. A poset P having countably directed joins is called a countably approximating poset if for each x P, the set c x is countably directed and x = c x. A countably approximating poset which is also a complete lattice is called a countably approximating lattice. In a poset P, it is clear that x c y implies that x y. Since every countably directed set is directed, we have that x y implies x c y for all x, y P. In other words, we have y c y for each y P. However, the following example shows that the reverse implication need not be true. Example 2.3. Let P be the unit interval [0, 1]. For all x, y [0, 1], it is easy to check that x c y x y and that x y x =0=y or x<y. It is clear that every countable poset is a countably approximating poset since every countably directed subset of a countable poset has a maximal element. Proposition 2.4. Let P be a poset and S a countable subset of P such that S exists. If s c x for all s S, then S c x. Proof. Straightforward. By Proposition 2.4, in a complete lattice P, the set c x is automatically countably directed for each x P. So, a complete lattice P is countably approximating iff for each x P, x = c x. Thus every continuous lattice is a countably approximating lattice. Definition 2.5. (see [9]) Let P be a poset. A subset X is said to have multiple joins if for all y ub(x), there is a unique x mub(x) with x y. In a poset P, it is clear that the empty set has multiple joins iff P has a bottom.

4 450 Xuxin Mao and Luoshan Xu Lemma 2.6. Let P be a poset, x P and ϕ = x. Then for A ϕ = x, ϕ A = A whenever A exists, where ϕ A denotes the supremum of the subset A in the principal ideal ϕ = x. Proof. Straightforward. 3 Principal ideal countably approximating posets In this section, we introduce the concept of principal ideal countably approximating posets and show that the principal ideal countably approximating posets are exactly the countably approximating posets. Definition 3.1. Let P be a poset having countably directed joins. If every principal ideal of P is countably approximating, then P is called a principal ideal countably approximating poset. Proposition 3.2. Let P be a countably approximating poset. Then P is a principal ideal countably approximating poset. Furthermore, x P, u ϕ = x, we have c u = c,ϕ u, where c,ϕ u is the set of all elements that are countably way-below u in the principal ideal ϕ, i.e., c,ϕ u = {y ϕ y c,ϕ u}. Proof. It suffices to show that x P, u ϕ = x, the equation c u = c,ϕ u holds. By Lemma 2.6, it is easy to check that c u c,ϕ u. Suppose that y c,ϕ u. By the countably approximating property of P and Lemma 2.6, the set c u c,ϕ u is countably directed and u = c u = ϕ c u. So there is v c u such that y v c u. Hence y c u and c,ϕ u c u. This shows that c u = c,ϕ u, as desired. Proposition 3.3. If P is a principal ideal countably approximating poset, then P is a countably approximating poset. Proof. Let P be a principal ideal countably approximating poset. Then for all x P, the principal ideal ϕ = x is a countably approximating poset. We claim that c x = c,ϕ x. By Lemma 2.6, it is easy to check that c x c,ϕ x. Suppose that y c,ϕ x. Let D P be a countably directed subset with h := D x, then the principal ideal ψ = h is a countably approximating poset. Thus x is the countably directed supremum of elements that are countably way-below it in the poset h, i.e., x = ψ c,ψ x = ψ {t ψ t c,ψ x}. By Lemma 2.6, x = ψ c,ψ x = ϕ c,ψ x. It follows from y c,ϕ x that there exists t c,ψ x such that y t c,ψ x. Since x D = ψ D, there is some d D such that y t d. This shows that y c x and c,ϕ x c x. Thus for all x P,wehave c x = c,ϕ x. By Lemma 2.6 and the principal ideal countably approximating property of P, x = ϕ c,ϕ x = ϕ c x = c x. By Definition 2.2, P is a countably approximating poset.

5 Order-theoretical characterizations 451 By Propositions 3.2 and 3.3, we immediately have the following characterization of countably approximating posets by principal ideals. Theorem 3.4. A poset P having countably directed joins is a countably approximating poset if and only if P is a principal ideal countably approximating poset. 4 Closed interval countably approximating posets In this section, we will consider the closed interval countably approximating posets. Firstly we introduce the following concept. Definition 4.1. Let P be a poset having countably directed joins. If every closed interval [x, y] in P is a countably approximating poset (in the relative order), then P is called a closed interval countably approximating poset. Applying Theorem 3.4 to the principal filters of a poset, we immediately have Proposition 4.2. Let P be a poset having countably directed joins. Then P is a closed interval countably approximating poset iff each principal filer x is a countably approximating poset (in the relative order). Definition 4.3. Let P be a poset having countably directed joins and x P. If D c x is countably directed and sup D = x, then D is called a local c-basis of x. The proof of the following lemma is straightforward and omitted. Lemma 4.4. Let P be a poset having countably directed joins. Then P is a countably approximating poset if and only if for all x P, x has a local c-basis. Theorem 4.5. If P is a closed interval countably approximating poset satisfying the controlling condition from below that for all x P, c x, then P is a countably approximating poset. Proof. For all x P, pick u c x. Let θ =[u, x]. Since P is a closed interval countably approximating poset, the closed interval θ = [u, x] isa countably approximating poset. Thus the set c,θ x of all elements that are countably way-below x in θ is countably directed and x = θ c,θ x = c,θ x. We claim that c,θ x c x. Suppose v c,θ x. Then u v c,θ x. For any countably directed subset D in P such that x h := D, it follows from u c x that there exists d 0 D such that u d 0. Let D 0 = D d 0. It is easy to see that D 0 is countably directed and D 0 = D = h. Let ρ =[u, h]. Clearly D 0 ρ and D 0 = ρ D 0. Since the closed interval [u, h] is a countably

6 452 Xuxin Mao and Luoshan Xu approximating poset and [u, x] is a principal ideal of [u, h], by Proposition 3.2, we have c,θ x = c,ρ x. Thus v c,ρ x. Since v c,ρ x h = D 0 = ρ D 0, there is some d D such that v d. This shows that v c x and hence c,θ x c x. By Definition 4.3, c,θ x is a local c-basis of x. It follows from Lemma 4.4 that P is a countably approximating poset. Corollary 4.6. Every closed interval countably approximating poset with a bottom is a countably approximating poset. Proof. Applying Theorem 4.5 or Proposition 3.3. Theorem 4.7. Let P be a countably approximating poset. If every nonempty finite subset of P has multiple joins, then P is a closed interval countably approximating poset. Proof. For all x, y P with x y, let θ =[x, y]. For each z [x, y], by the countably approximating property of P, the set c z is countably directed and z = c z. For all u c z,wehave{x, u} y. By hypothesis there exists a unique t u mub{x, u} such that t u y. It is routine to check by the countably directedness of c z that the set {t u u c z} [x, y] is countably directed and that z = {t u u c z} = θ {t u u c z}. We claim that {t u u c z} c,θ z. For each t u, let D be any countably directed subset in [x, y] such that z θ D. Since u c z θ D = D, there exists some d D such that u d. This shows that {x, u} d y and hence t u d by the minimality of t u.thust u c,θ z and {t u u c z} c,θ z. By Definition 4.3, {t u u c z} is a local c-basis of z in [x, y]. By Lemma 4.4, [x, y] is a countably approximating poset. Hence P is a closed interval countably approximating poset. Corollary 4.8. Let P be a a poset having countably directed joins in which every finite subset has multiple joins. Then P has a bottom and P is a countably approximating poset iff P is a closed interval countably approximating poset. Proof. Applying Theorems 4.5 and 4.7. Lemma 4.9. Let P be a poset. If P has countably directed joins and countable joins, then P is a complete lattice. Proof. For any A P, let D = { S S is a countable subset of A}. It is easy to check that D is countably directed and D is the supremum of A. Corollary Let P be a countably approximating poset in which every nonempty countable subset has multiple joins. Then every closed interval of P is a countably approximating lattice.

7 Order-theoretical characterizations 453 Proof. By Theorem 4.7, it suffices to show that every closed interval θ =[x, y] of P is a complete lattice. Clearly, the closed interval [x, y] has a bottom x. Since every nonempty countable subset of P has multiple joins, every countable subset in [x, y] has a supremum. It follows from P has countably directed joins that every closed interval [x, y] has countably directed joins. Thus every closed interval [x, y] ofp has countably directed joins and countable joins. By Lemma 4.9, [x, y] is a complete lattice. Theorem Let P be a poset having countably directed joins. If every closed interval [x, y] of P is countable, then P is a countably approximating poset. Proof. Let D be any countably directed subset of P and t = D. Pick d D. Clearly t = D = (D d). By hypothesis the closed interval [d, t] is countable. Since D is countably directed, the set D d [d, t] is countably directed and hence has a maximal element. This shows that t D d D. Thus every countably directed subset of P has a maximal element. So, for all x P,wehave c x = x and P is a countably approximating poset. References [1] G. Gierz, K. Hofmann, K. Keimel, et al. Continuous Lattices and Domains. Cambridge University Press, Cambridge, [2] Y. H. Han, S. S. Hong, C. K. Lee, P. U. Park. A generalization of continuous posets. Comm. Korean Math. Soc., 1989, 4(1): [3] R. E. Hoffmann. Continuous posets and adjoint sequences. Semigroup Forum, 1979, 18: [4] J. D. Lawson. The duality of continuous posets. Houston J. of Mathematics, 1979, 5: [5] J. D. Lawson, Luoshan Xu. Posets Having Continuous Intervals. Theoretical Computer Science, 2004, 316: [6] S. O. Lee. On countably approximating lattices. J. Korean Math. Soc., 1988, 25(1): [7] Xuxin Mao, Luoshan Xu. Meet continuity properties of posets. Theoretical Computer Science, 2009, 410: [8] D. S. Scott. Continuous Lattice. Lecture Notes in Mathematics 274, Springer-Verlag, Berlin, 1972,

8 454 Xuxin Mao and Luoshan Xu [9] V. Stolenberg-Hansen, I. Lindstrom and E. R. Griffor. Mathematical theory of domains. Cambridge University Press, Cambridge, [10] Luoshan Xu, Closed interval continuity and principal ideal continuity on posets, J. of Yangzhou University (Nature Science Edition), 1999, 2(4): 1-5. (in Chinese) [11] Luoshan Xu. Continuity of Posets via Scott Topology and Sobrification. Topology and Its Applications, 2006, 153: Received June 7, 2014