Varieties of complex algebras

Goldblatt, Robert

doi:10.1016/0168-0072(89)90032-8

Cited by 231 publications

(252 citation statements)

References 26 publications

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“…A detailed proof for general pre-orders can be found in [25]. The technique is classical in mathematics; for related results see among others [14,19,15] (and also [7] for a survey).…”

Section: Elementwise Liftingmentioning

confidence: 99%

A Discrete Geometric Model of Concurrent Program Execution

Möller

Hoare

Müller

et al. 2017

Unifying Theories of Programming

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Abstract. A trace of the execution of a concurrent object-oriented program can be displayed in two-dimensions as a diagram of a non-metric finite geometry. The actions of a programs are represented by points, its objects and threads by vertical lines, its transactions by horizontal lines, its communications and resource sharing by sloping arrows, and its partial traces by rectangular figures. We prove informally that the geometry satisfies the laws of Concurrent Kleene Algebra (CKA); these describe and justify the interleaved implementation of multithreaded programs on computer systems with a lesser number of concurrent processors. More familiar forms of semantics (e.g., verification-oriented and operational) can be derived from CKA. Programs are represented as sets of all their possible traces of execution, and non-determinism is introduced as union of these sets. The geometry is extended to multiple levels of abstraction and granularity; a method call at a higher level can be modelled by a specification of the method body, which is implemented at a lower level. The final section describes how the axioms and definitions of the geometry have been encoded in the interactive proof tool Isabelle, and reports on progress towards automatic checking of the proofs in the paper.

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“…A detailed proof for general pre-orders can be found in [25]. The technique is classical in mathematics; for related results see among others [14,19,15] (and also [7] for a survey).…”

Section: Elementwise Liftingmentioning

confidence: 99%

A Discrete Geometric Model of Concurrent Program Execution

Möller

Hoare

Müller

et al. 2017

Unifying Theories of Programming

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“…Altogether this yields what is known as extended Stone or Priestley duality. The details for a fairly large class of additional operations may be found in the first section of [8].…”

Section: Duality For Additional Operationsmentioning

confidence: 99%

“…This principle was applied first by Stone himself in functional analysis, followed by Grothendieck in algebraic geometry who represented rings in terms of sheaves over the dual spaces of distributive lattices (i.e., 'positive' Boolean algebras) and has since, over and over again, proved itself central in logic and its applications in computer science. One may specifically mention Scott's model of the λ-calculus, which is a dual space, Esakia's duality [4] for Heyting algebras and the corresponding frame semantics for intuitionist logics, Goldblatt's paper [8] identifying extended Stone duality as the theory for completeness issues for Kripke semantics in modal logic, and Abramsky's path-breaking paper [1] linking program logic and domain theory. Our work with Grigorieff and Pin [7,9,6], with Pippenger [10] as a precursor, shows that the connection between regular languages and monoids also is a case of Stone duality.…”

Section: Stone Dualitymentioning

confidence: 99%

Duality and Recognition

Gehrke

2011

Mathematical Foundations of Computer Science 2011

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Abstract. The fact that one can associate a finite monoid with universal properties to each language recognised by an automaton is central to the solution of many practical and theoretical problems in automata theory. It is particularly useful, via the advanced theory initiated by Eilenberg and Reiterman, in separating various complexity classes and, in some cases it leads to decidability of such classes. In joint work with Jean-Éric Pin and Serge Grigorieff we have shown that this theory may be seen as a special case of Stone duality for Boolean algebras extended to a duality between Boolean algebras with additional operations and Stone spaces equipped with Kripke style relations. This is a duality which also plays a fundamental role in other parts of the foundations of computer science, including in modal logic and in domain theory. In this talk I will give a general introduction to Stone duality and explain what this has to do with the connection between regular languages and monoids. Stone dualityStone type dualities is the fundamental tool for moving between linguistic specification and spatial dynamics or transitional unfolding. As such, it should come as no surprise that it is a theory of central importance in the foundations of computer science where one necessarily is dealing with syntactic specifications and their effect on physical computing systems. In 1936, M. H. Stone initiated duality theory by presenting what, in modern terms, is a dual equivalence between the category of Boolean algebras and the category of compact Hausdorff spaces having a basis of clopen sets, so-called Boolean spaces [13]. The points of the space corresponding to a given Boolean algebra are not in general elements of the algebra -just like states of a system are not in general available as entities in a specification language but are of an entirely different sort. In models of computation these two different sorts, specification expressions and states, are given a priori but in unrelated settings. Via Stone duality, the points of the space may be obtained from the algebra as homomorphisms into the two-element Boolean algebra or equivalently as ultrafilters of the algebra. In logical terms these are valuations or models of the Boolean algebra. In computational terms they are possible states of the system. Each element of the Boolean algebra corresponds to the set of all models in which it is true, or all states in which it holds, and the topology of the space is generated by these sets. A main insight of Stone is that one may recover the original algebra as the Boolean algebra of clopen subsets of the resulting space.

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“…Because of this, Esakia relations are also called continuous relations. (2) It is easy to see that Esakia relations are exactly the inverses of binary JT-relations with the same source and target (see, e.g., [21]). Inverses of binary JT-relations with not necessarily the same source and target were first studied by Halmos [22].…”

Section: Remark 42mentioning

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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Varieties of complex algebras

Cited by 231 publications

References 26 publications

A Discrete Geometric Model of Concurrent Program Execution

A Discrete Geometric Model of Concurrent Program Execution

Duality and Recognition

Irreducible Equivalence Relations, Gleason Spaces, and de Vries Duality

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