Continuity in Partially Ordered Sets

The notion of a continuous domain is generalized to include posets which are not dcpos and in which the set of elements way below an element is not necessarily directed. We show that several of the pleasing algebraic and topological properties of domains carry over to this setting.


Introduction
Continuous lattices and their generalizations, continuous domains, have been studied for more than three decades.Continuous lattices are complete lattices where each element is the supremum of elements way below it.For a poset to be a continuous domain, it needs to have sups of only directed sets in addition to the following two conditions: i each element is the sup of elements way below it, and ii for each element, the set of elements way below it is a directed set.A continuous poset 1 is any poset in which the conditions i and ii are satisfied.In a complete lattice, in fact in any sup-semilattice, the condition ii above is automatically satisfied.In 2 , the authors have studied lattices which are not complete but satisfy the conditions i and ii above.The purpose of this paper is to study posets which need not be dcpos or lattices but which satisfy the condition that each element is the sup of elements way below it.The exact definition will be given in Section 2.
Here we recall some basic definitions and terminology from domain theory; more details can be found in 1 .For x, y ∈ P, a partially ordered set, we say that x is way below y written x y , if whenever y ≤ sup D, for a directed set D, there exists d ∈ D such that x ≤ d.A continuous poset is a partially ordered set P in which the following two conditions are satisfied.i For each x ∈ P , x sup {y : y x}; and ii for each x ∈ P , {y : y x} is a directed set.A continuous poset in which every directed set has a least upper bound such posets are called dcpos is called a continuous domain.A continuous poset which is also a complete lattice is called a continuous lattice.In a complete lattice, since the second condition above is automatically International Journal of Mathematics and Mathematical Sciences satisfied, one needs to verify only the first condition.In any continuous poset P, the way below relation has the interpolation property, that is, for x, y ∈ P with x y, there exists an element z ∈ P, such that x z y.For more information about continuous lattices, and continuous domains, the standard reference is 1 .A subset S of a poset P is called up-complete if for any directed subset D of S, for which sup D exists in P , sup D is contained in S. We use the following notations: ↓ A {x : x ≤ a for some a ∈ A}; ⇓ A {x : x a for some a ∈ A}; and ⇓ x ⇓ {x}.A subset J of a poset P is called a lower set if and only if J ↓ J. Recall that a subset U of a poset P is Scott open if and only if U is an upper set satisfying the property that if the supremum of any directed set D is in U, then D itself intersects U.The lower topology on a poset P has subbasic closed sets of the form ↑ x, x ∈ P. The join of the lower topology and the Scott topology on P is the Lawson topology and is denoted by λ P .

C-posets
The following simple but crucial result is the motivation for our definition of a C-poset.
Theorem 2.1.Let L be a continuous poset.Then for each x, y ∈ L, with x y, there exists u ∈ L and an up-complete lower set J ⊆ P such that u y, x / ∈ J, and ↑ u ∪ J L.
Proof.Suppose that L is a continuous poset.Let x, y ∈ L with x y.Then ∃u x with u y.
sup D, then there exists w such that v w sup D. This implies that ∃d ∈ D such that v w ≤ d, a contradiction.This shows that the supremum of D belongs to J, and thus J is up-complete.It is immediate that u y, x / ∈ J, and ↑ u ∪ J L. Definition 2.2.A partially ordered set P is called a C-poset if for any x, y ∈ P, with x y, ∃ u y and an up-complete lower set J such that x / ∈ J, and ↑ u ∪ J P. Example 2.3.i If S is an infinite set and if P is the poset of all finite and cofinite subsets of S, then P is a C-poset.Suppose that A, B are in P such that A B. Let x ∈ A such that x / ∈ B. If we define J {A : A ∈ P, x / ∈ A}, and u {x}, then this J and u satisfy the conditions in the definition of a C-poset.
ii Let X be a T 0 space for which the lattice of open sets is a continuous lattice.Here X is endowed with the specialization order.Let L be a continuous domain with a least element endowed with the Scott topology.Let P X → L be the set of all Scott continuous functions.It can be shown that P is a C-poset but not a continuous poset, see 1, page 200 .
iii Let P i , i ∈ I be a collection of C-posets with 0 and 1.Then the product of P i s is a Cposet.If x i y i , then there exists j ∈ I such that x j y j .Since P j is a C-poset, there exists a ∈ P j such that a y j and an up-complete lower set K of P j such that x j / ∈ K and P j ↑ a ∪ K. Define u u i , where u i 0 if i / j and u j a.Also define J ↓ S, where S { x i : x i 1 if i / j and x j ∈ K}.It is easy to see that the above u and J satisfy the conditions in the definition of a C-poset.
The next theorem establishes that C-posets do satisfy the approximation property, that is, each element is the supremum of elements way below it.
Theorem 2.4.If P is a C-poset, then for each x ∈ P , x sup {z : z x} sup ⇓ x.
Proof.We need to prove this only for x / 0. If x / 0, then there exists y ∈ P such that x y.Therefore, by the definition of a C-poset, there exists u y, and an up-complete lower set J such that x / ∈ J, and ↑ u ∪ J P .We will show that u x.Suppose that D is a directed set in P such that sup D ≥ x.If D ∩ ↑ u ∅, then D ⊆ J and since J is up-complete, sup D ∈ J. Since J is a lower set, this forces x ∈ J, a contradiction.This shows that u x.Thus the set {z : z x} is nonempty, and x is an upper bound of it.Let w be any upper bound of that set.We will show that x ≤ w.Indeed, if x w, then ∃u w and an up-complete lower set J such that x / ∈ J , and ↑ u ∪ J P. But then u x, which contradicts the assumption that w is an upper bound of the set {z : z x}.Therefore x ≤ w.This completes the proof of the theorem.

Corollary 2.5. If a C-poset is a sup-semilattice, then it is a continuous poset; if a C-poset is a complete lattice, then it is a continuous lattice.
Theorem 2.6.Let P be a C-poset, and let {x i,k : i ∈ I, k ∈ K i } be a nonempty family of elements of P such that {x i,k : k ∈ K i } is a directed set for all i ∈ I. Then the following identity holds whenever the specified sups and infs exist: where M is the set of all choice functions f : Proof.Since the right hand side is always less than or equal to the left-hand side, we need to prove only the reverse inequality.By Theorem 2.4, it is enough to show that whenever x i∈I K i x i,k , we have x f∈M i∈I x i,f i .Suppose that x i∈I K i x i,k .Then, x K i x i,k , for all i ∈ I. Therefore, by the definition of the way below relation, we can choose a g i ∈ K i with x ≤ x i,g i for all i ∈ I.This implies that x ≤ i∈I x i,g i , and hence x is less than or equal to the right hand side of the identity.This completes the proof of the theorem.
Let g : P → Q, and let d : Q → P be monotone functions between posets.The pair g, d is called a Galois connection if for x ∈ P, and y ∈ Q, g x ≥ y if and only if x ≥ d y .Here g is called an upper adjoint, and d is called a lower adjoint.A monotone function g : P → Q between posets is called an upper adjoint if there exists a necessarily unique monotone function d : Q → P such that g, d is a Galois connection.Basic properties of Galois connections can be found in 1 .
Definition 2.7.A function from a C-poset to another C-poset is called a homomorphism if it is an upper adjoint which preserves sups of directed sets.A subposet S of a C-poset P is called a subalgebra if the inclusion map i : S → P is a homomorphism.Theorem 2.8. a Let P be a C-poset and let Q be any poset.
Proof. a Suppose that g is the lower adjoint of f.Let y 2 y 1 in Q. Pick x 1 ∈ P with f x 1 y 1 .Since g is a lower adjoint of f, y 2 f x 1 implies g y 2 x 1 .Then there exist u ∈ P and an up-complete lower set J such that u x 1 , g y 2 / ∈ J, and P ↑ u ∪ J. Define I : g −1 J , and v : f u .Since the lower adjoint g preserves sups, I is up-complete, and b Let i : S → P be the inclusion map and let d be its lower adjoint.Let x, y ∈ S such that x y.Then there exists u ∈ P, and an up-complete lower set J ⊆ P, such that u y, x / ∈ J, and P ↑ u ∪ J. Since u y i y , d u y.Define I : i −1 J .Since i preserves sups of directed sets, I is an up-complete lower set, and x / ∈ I.For t ∈ S, t / ∈ J, implies i t t / ∈ J, which implies that u ≤ t.Thus d u ≤ d t t.Therefore, ↑ S d u ∪ I S.This completes the proof that S is a subalgebra.
Notice that the proof of a above did not use the assumption that f preserves the sups of directed sets.Definition 2.9.Let P be a C-poset.A subposet B of P is called a basis of P if given any x, y ∈ P with x y, there exist b ∈ B and an up-complete lower set J ⊆ P such that b y, x / ∈ J, and ↑ b ∪ J P .Proof. 1 Let d be the lower adjoint of g.Suppose that y 1 , y 2 ∈ Q such that y 2 y 1 .Let x 1 ∈ P, such that g x 1 y 1 .Since y 2 g x 1 , d y 2 x 1 .Since B is a basis of the C-poset P, there exist b ∈ B and an up-complete lower set J ⊆ P such that b x 1 , d y 2 / ∈ J, and ↑ b ∪ J P. Let I d −1 J and c f b .Since the lower adjoint preserves sups, I is an up-complete lower set, and y 2 / ∈ I.If x ∈ J, f x ≤ f x implies df x ≤ x.This implies that df x ∈ J, and hence The proof of 2 is similar and hence omitted.Though algebraic lattices were studied for several decades before the introduction of continuous lattices, algebraic lattices, and their generalizations, algebraic domains, have played an important role in domain theory.Algebraic domains are continuous domains in which every element is the sup of compact elements an element x is compact if x x below it, in which the set of compact elements below every element is a directed set.We generalize this notion to posets which are not necessarily dcpos in which the set of compact elements below any element is not necessarily directed.The following result is the motivation for our definition.

Theorem 2 .
10.If B is a basis of a C-poset P, then for each x ∈ P , x sup {b : b ∈ B, b x}.Proof.We need to prove this only for x / 0. If x / 0, then there exists y ∈ P such that x y.Then, by the definition of the basis, there exists b ∈ B such that b y, and an up-complete lower set J such that x / ∈ J, and ↑ b ∪ J P. It follows immediately that b x.Thus, {b : b ∈ B, b x} is nonempty and x is an upper bound of it.Suppose that v is any upper bound of the same set.If x v, then there exist b ∈ B and an up-complete lower set J such that x / ∈ J , and ↑ u ∪ J P. As shown above b x which contradicts the assumption that v is an upper bound of {b : b ∈ B, b x}.Thus x ≤ u.This completes the proof.Proposition 2.11. 1 Let P, Q be C-posets with B as a basis of P .If g : P → Q is a surjective homomorphism, then g B is a basis of Q. 2 If S is a subalgebra of a C-poset P, and if B is a basis of P, then d B is a basis of S, where d is the lower adjoint of the inclusion map.

D 2 {d 2 :
∃d 1 such that d 1 , d 2 ∈ D}.Then D 1 and D 2 are directed sets such that sup D 1 a, and sup D 2 b.Since P is meet-continuous by Proposition 3.2, a ∧ b sup D 1 ∧ sup D 2 sup D 1 D 2 ∈ O. Since D 1 D 2 is a directed set and since O is Scott-open, there exist d 1 ∈ D 1 and d 2 ∈ D 2 such that d 1 ∧ d 2 ∈ O, which implies that d 1 , d 2 ∈ U.This completes the proof of the proposition.