A partially ordered space which is not a priestley space

Stralka, Albert

doi:10.1007/bf02572690

Cited by 25 publications

(16 citation statements)

References 2 publications

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“…This is not the case for every choice of M ∼ . A classic example of Stralka [40] shows that the ordered space M ∼ = {0, 1}; ≤, T , which generates the category of Priestley spaces, is not standard.…”

Section: Background Motivation and Overview Of Resultsmentioning

confidence: 99%

Standard topological algebras: syntactic and principal congruences and profiniteness

et al. 2005

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A topological quasi-variety Q + T (M ∼ ) := IScP + M ∼ generated by a finite algebra M ∼with the discrete topology is said to be standard if it admits a canonical axiomatic description. Drawing on the formal language notion of syntactic congruences, we prove that Q + T (M ∼ ) is standard provided that the algebraic quasi-variety generated by M ∼ is a variety, and that syntactic congruences in that variety are determined by a finite set of terms. We give equivalent semantic and syntactic conditions for a variety to have Finitely Determined Syntactic Congruences (FDSC), show that FDSC is equivalent to a natural generalisation of Definable Principle Congruences (DPC) which we call Term Finite Principle Congruences (TFPC), and exhibit many familiar algebras M ∼ that our method reveals to be standard. As an application of our results we show, for example, that every Boolean topological lattice belonging to a finitely generated variety of lattices is profinite and that every Boolean topological group, semigroup, and ring is profinite. While the latter results are well known, the result on lattices was previously known only in the distributive case. Background, motivation and overview of resultsAn algebra M = M ; F with finite underlying set M and operations F generates an (algebraic) quasi-variety Q(M) := ISP M consisting of all isomorphic copies of subalgebras of direct powers of M. Similarly a structure M ∼ = M ; G, H, R, T with finite underlying set M , operations G, partial operations H, relations R and discrete topology T generates a topological quasi-variety Q + T (M ∼ ) := IS c P + M ∼ consisting of all isomorphic copies of topologically closed substructures of non-zero direct powers, with the product topology, of M ∼ . Interest in topological quasi-varieties stems from the fact that they arise as the duals to algebraic quasi-varieties under natural dualities. The general theory of natural dualities provides methods to Presented by R. W. Quackenbush.

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Section: Background Motivation and Overview Of Resultsmentioning

confidence: 99%

Standard topological algebras: syntactic and principal congruences and profiniteness

et al. 2005

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“…If R satisfies the Priestley separation axiom, then we call R a Priestley quasi-order. Each Priestley quasi-order is closed, but the converse is not true in general [10,30]. A quasiordered Priestley space is a pair (X, R), where X is a Stone space and R is a Priestley quasi-order on X.…”

Section: Lattice Subordinations and The Priestley Separation Axiommentioning

confidence: 99%

Irreducible Equivalence Relations, Gleason Spaces, and de Vries Duality

Bezhanishvili

Sourabh

et al. 2016

Appl Categor Struct

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By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras (complete Boolean algebras endowed with a proximity-like relation). We provide an alternative "modal-like" duality by introducing the concept of a Gleason space, which is a pair (X, R), where X is an extremally disconnected compact Hausdorff space and R is an irreducible equivalence relation on X. Our main result states that the category of Gleason spaces is equivalent to the category of compact Hausdorff spaces, and is dually equivalent to the category of de Vries algebras.

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“…Indeed, Stralka's well-known example (which we discuss further in Sect. 5.3) shows that the class of Priestley spaces (which is the Boolean core P BC of the class of ordered sets P) is not equal to the class of Boolean topological ordered sets [14]. Thus P is not standard.…”

Section: Relational Structuresmentioning

confidence: 99%

Boolean topological graphs of semigroups: the lack of first-order axiomatization

Stronkowski

Trotta²

2014

Semigroup Forum

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The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. For a class C of algebras letAssume that C is a class of semigroups possessing a nontrivial member with a neutral element and let H be the universal Horn class generated by G(C ). We prove that the Boolean core of H , i.e., the topological prevariety generated by finite members of H equipped with the discrete topology, does not admit a first-order axiomatization relative to the class of all Boolean topological structures in the language of H . We derive analogous results when C is a class of monoids or groups with a nontrivial member.

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A partially ordered space which is not a priestley space

Cited by 25 publications

References 2 publications

Standard topological algebras: syntactic and principal congruences and profiniteness

Standard topological algebras: syntactic and principal congruences and profiniteness

Irreducible Equivalence Relations, Gleason Spaces, and de Vries Duality

Boolean topological graphs of semigroups: the lack of first-order axiomatization

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