Bisimulations for coalgebras on Stone spaces

We introduce pairwise Stone spaces as a bitopological generalisation of Stone spaces – the duals of Boolean algebras – and show that they are exactly the bitopological duals of bounded distributive lattices. The category PStone of pairwise Stone spaces is isomorphic to the category Spec of spectral spaces and to the category Pries of Priestley spaces. In fact, the isomorphism of Spec and Pries is most naturally seen through PStone by first establishing that Pries is isomorphic to PStone, and then showing that PStone is isomorphic to Spec. We provide the bitopological and spectral descriptions of many algebraic concepts important in the study of distributive lattices. We also give new bitopological and spectral dualities for Heyting algebras, thereby providing two new alternatives to Esakia's duality.

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Measurable refinement monoids and applications to distributive semilattices, Heyting algebras, and stone spaces

Mathematische Zeitschrift ◽

10.1007/bf01163161 ◽

1984 ◽

Vol 187 (1) ◽

pp. 13-21 ◽

Cited By ~ 4

Author(s):

Hans Dobbertin

Keyword(s):

Heyting Algebras ◽

Stone Spaces ◽

Distributive Semilattices ◽

Refinement Monoids

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STONE SPACES (Cambridge Studies in Advanced Mathematics 3)

Bulletin of the London Mathematical Society ◽

10.1112/blms/19.2.195b ◽

1987 ◽

Vol 19 (2) ◽

pp. 195-197

Author(s):

J. L. Bell

Keyword(s):

Advanced Mathematics ◽

Stone Spaces

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Stone spaces and compactifications

Pure Mathematical Sciences ◽

10.12988/pms.2013.13010 ◽

2013 ◽

Vol 2 ◽

pp. 75-81

Author(s):

M. Al-Hajri ◽

K. Belaid ◽

O. Echi

Keyword(s):

Stone Spaces

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TWO SPACES OF MINIMAL PRIMES

Journal of Algebra and Its Applications ◽

10.1142/s0219498811005373 ◽

2012 ◽

Vol 11 (01) ◽

pp. 1250014 ◽

Cited By ~ 4

Author(s):

PAPIYA BHATTACHARJEE

Keyword(s):

Compact Space ◽

Hausdorff Space ◽

Topological Spaces ◽

Zariski Topology ◽

Topological Properties ◽

Locally Compact ◽

Inverse Topology ◽

Minimal Prime ◽

Stone Spaces ◽

Prime Elements

This paper studies algebraic frames L and the set Min (L) of minimal prime elements of L. We will endow the set Min (L) with two well-known topologies, known as the Hull-kernel (or Zariski) topology and the inverse topology, and discuss several properties of these two spaces. It will be shown that Min (L) endowed with the Hull-kernel topology is a zero-dimensional, Hausdorff space; whereas, Min (L) endowed with the inverse topology is a T1, compact space. The main goal will be to find conditions on L for the spaces Min (L) and Min (L)-1 to have various topological properties; for example, compact, locally compact, Hausdorff, zero-dimensional, and extremally disconnected. We will also discuss when the two topological spaces are Boolean and Stone spaces.

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Types and Stone spaces

10.1093/oso/9780198790396.003.0014 ◽

2018 ◽

Author(s):

Tim Button ◽

Sean Walsh

Keyword(s):

Topological Space ◽

Model Theory ◽

Algebraic Structure ◽

Possible Worlds ◽

Elementary Extension ◽

Philosophical Significance ◽

Related Algebra ◽

Contemporary Study ◽

Stone Spaces

Types are one of the cornerstones of contemporary model theory. Simply put, a type is the collection of formulas satisfied by an element of some elementary extension. The types can be organised in an algebraic structure known as a Lindenbaum algebra. But the contemporary study of types also treats them as the points of a certain kind of topological space. These spaces, called ‘Stone spaces’, illustrate the richness of moving back-and-forth between algebraic and topological perspectives. Further, one of the most central notions of contemporary model theory—namely stability—is simply a constraint on the cardinality of these spaces. We close the chapter by discussing a related algebra-topology ‘duality’ from metaphysics, concerning whether to treat propositions as sets of possible worlds or vice-versa. We show that suitable regimentations of these two rival metaphysical approaches are biinterpretable (in the sense of chapter 5), and discuss the philosophical significance of this rapprochement.

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