Priestley space

In mathematics, a Priestley space is an ordered topological space with special properties. Priestley spaces are named after Hilary Priestley who introduced and investigated them. Priestley spaces play a fundamental role in the study of distributive lattices. In particular, there is a duality ("Priestley duality") between the of Priestley spaces and the category of bounded distributive lattices. A Priestley space is an ordered topological space (X,τ,≤), i.e. a set X equipped with a partial order ≤ and a topology τ, satisfying the following two conditions: (X,τ) is compact. If , then there exists a clopen up-set U of X such that x∈U and y∉ U. (This condition is known as the Priestley separation axiom.) Each Priestley space is Hausdorff. Indeed, given two points x,y of a Priestley space (X,τ,≤), if x≠ y, then as ≤ is a partial order, either or . Assuming, without loss of generality, that , (ii) provides a clopen up-set U of X such that x∈ U and y∉ U. Therefore, U and V = X − U are disjoint open subsets of X separating x and y. Each Priestley space is also zero-dimensional; that is, each open neighborhood U of a point x of a Priestley space (X,τ,≤) contains a clopen neighborhood C of x. To see this, one proceeds as follows. For each y ∈ X − U, either or . By the Priestley separation axiom, there exists a clopen up-set or a clopen down-set containing x and missing y. The intersection of these clopen neighborhoods of x does not meet X − U. Therefore, as X is compact, there exists a finite intersection of these clopen neighborhoods of x missing X − U. This finite intersection is the desired clopen neighborhood C of x contained in U. It follows that for each Priestley space (X,τ,≤), the topological space (X,τ) is a Stone space; that is, it is a compact Hausdorff zero-dimensional space. Some further useful properties of Priestley spaces are listed below. Let (X,τ,≤) be a Priestley space. (a) For each closed subset F of X, both ↑ F = {x ∈ X : y ≤ x for some y ∈ F} and ↓ F = { x ∈ X : x ≤ y for some y ∈ F} are closed subsets of X.

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Concepts associés (3)

Birkhoff's representation theorem

This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation). In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders.

Duality theory for distributive lattices

In mathematics, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces. This duality, which is originally also due to Marshall H. Stone, generalizes the well-known Stone duality between Stone spaces and Boolean algebras. Let L be a bounded distributive lattice, and let X denote the set of prime filters of L. For each a ∈ L, let φ+(a) = {x∈ X : a ∈ x.

Distributive lattice

In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets. As in the case of arbitrary lattices, one can choose to consider a distributive lattice L either as a structure of order theory or of universal algebra.