Stone, M. H. (1937). Topological representations of distributive lattices and Brouwerian logics.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....: a . Theorem 2.1. 7, 12] The Stone map s is a reduced and perfect representation of B. Conversely, if e : B is a reduced and perfect representation of B, there is a unique bijection j : X lt(B) such that e(b) j 1 [s(b) The following result is a variant of Stone s Separation Lemma [13]: Lemma 2.2. If I is an ideal of B and F a filter such that I = 0, then there is some F I = 0. We will frequently use the finite co finite algebra FC(w) of the set w of natural numbers as a source of examples. FC(w) is the subalgebra of 2 w consisting of all finite and all ....

Stone, M. (1937). Topological representations of distributive lattices and Brouwerian logics. Casopis Pest. Mat., 67, 1--25.

.... normal operators (section 7, part II) Representation and duality for algebras relies on having an appropriate representation theorem for general lattices, just like the tradition of similar theorems started in [30] relies on the Stone [41] representation of Boolean algebras and the Stone [42] or the Priestley [37] representation of distributive lattices. Part I is taken up with establishing a simple solution to this problem . Our lattice representation is inspired by and abstracts over both Goldblatt s [17] and Urquhart s [46] It is based on the simple idea that since meets and ....

M. H. STONE (1937) "Topological Representation of Distributive Lattices and Brouwerian Logics", Casopsis pro Pestovani Matematiky a Fysiky 67, 1-25.

....This algebra bears a strong resemblance to the algebra of intuitionistic logic, in fact, it is two copies of a Heyting algebra, one order reversed, producted together. Thus it clearly contains two copies of intuitionistic logic (or a copy of intuitionistic logic and a copy of Browerian logic (Stone 1937)) We now show how these two copies can be extracted from constructible duality. Then we consider Girard s linear logic and in particular we discuss its exponentials which will be the tools that carry out our promised decomposition of N Gamma c . These exponentials allow us to provide a ....

Stone, M. 1937. Topological representations of distributive lattices and brouwerian logics.

....topological interior, satisfying (A B) A B and 1 = 1. Traditional intuitionistic logic is necessarily of this static kind since it lacks a separate dynamic intersection distinct from static intersection, cf. Stone s topological treatment of intuitionistic logic at the end of his paper [Sto37]. We define the dual notion of closure, A, as ( A ) and static implication A)B as A GammaffiB. We have seen that the language of linear logic is that of quantum logic expanded with dynamic connectives and constants, and A and A. The laws of linear logic, at least those of its ....

....will be a set of events ordered by time. State spaces generalize bc domains (Scott domains or bounded complete algebraic cpo s) Gun92] while their complementary event spaces generalize Winskel s event structures [Win86] This particular duality is one small fragment of Birkhoff Stone duality [Bir33, Sto36, Sto37, Pri70], with the partial distributive lattices alluded to in the section on measurement constituting a much larger fragment. A state space can be thought of as a representation of behavior somewhere in between a formal language 3 We shall see later that the terms dual, complement, and converse form a ....

M. Stone. Topological representations of distributive lattices and brouwerian logics. Casopis Pest. Math., 67:1--25, 1937.

....Here we limit the relations between events to those of temporal order ( binary distance ) and conflict. In the absence of conflict this arithmetic has for its progenitor the Birkhoff arithmetic of partial orders [Bir42] as a basic programming language for schedules. The Birkhoff Stone duality [Bir33, Sto37] of posets as deterministic schedules and distributive lattices as deterministic automata then constitutes a logic in which the duality of true and false are replaced by the duality of doing and being. This duality interchanges events (as actions) and states (as facts or models) and dualizes the ....

....infinite dualities we will use only the 6 bit duality numbering. The informal notion of infinite meets and joins is expressed formally as the Stone topology. In the case of posets and distributive lattices, putting the Stone topology on the left, duality 48, yields what Stone described obscurely [Sto37] but which was much later described more naturally by Priestley [Pri70] as ordered Stone spaces. In the special case where the order is discrete (but still duality 48) we have Stone spaces on the left and Boolean algebras on the right. When we add the Stone topology to the bounded distributive ....

M. Stone. Topological representations of distributive lattices and brouwerian logics. Casopis Pest. Math., 67:1--25, 1937.

.... normal operators (section 7, part II) Representation and duality for algebras relies on having an appropriate representation theorem for general lattices, just like the tradition of similar theorems started in [30] relies on the Stone [41] representation of Boolean algebras and the Stone [42] or the Priestley [37] representation of distributive lattices. Part I is taken up with establishing a simple solution to this problem 4 . Our lattice representation is inspired by and abstracts over both Goldblatt s [17] and Urquhart s [46] It is based on the simple idea that since meets and ....

M. H. STONE (1937) "Topological Representation of Distributive Lattices and Brouwerian Logics", Casopsis pro Pestovani Matematiky a Fysiky 67, 1-25.

....isomorphism of Fpos with itself and of FDL with itself. Dual continues to be defined as before: if P is a finite poset, 2 P is a finite distributive lattice, and vice versa, making Fpos dual to FDL (isomorphic to FDL op ) just as for Fchn and Fchn01, shown by Birkhoff in 1933 [Bir33] Stone [Sto37] a little later found one extremal extension of this duality to infinite objects, much later characterized nicely by Priestley [Pri70] as the duality of partially ordered Stone spaces and distributive lattices; the other extremal extension is between posets and profinite distributive lattices, for ....

M. Stone. Topological representations of distributive lattices and brouwerian logics. Casopis Pest. Math., 67:1--25, 1937.

....That where distribution is assumed and that where an orthonegation operator is present. In both cases the semantic analysis of logical operators can be based on a representation by sets of the Lindenbaum algebras: Boolean and distributive lattices or ortholattices (Goldblatt 1975, Priestley 1970, Stone 1937, Stone 1938) Propositions, the semantic counterparts of sentences, are modelled as certain kinds of sets (just any subsets of a fixed set for the distributive case, or only the regular subsets in the case of orthologic (Goldblatt 1974, 1975) What we intend to do in this paper is to extend this ....

Stone, Marshall H.: Topological Representation of Distributive Lattices and Brouwerian Logics. Casopsis pro Pestovani Matematiky a Fysiky 67 (1937), 1-25.

....by compiling such a schedule will now be a proper Boolean subalgebra of the power set of the set of all jobs in the schedule, due to the schedule restrictions eliminating some states, e.g. those containing only finitely many jobs. Stone did extend his Boolean duality to distributive lattices [Sto37], but purely topologically rather than via the more natural blend of order and topology devised by Priestley. As Rota put it, Stone s representation theorem of 1936 for distributive lattices closely imitated his representation theorem for Boolean algebras, and as a consequence turned out to be ....

M. Stone. Topological representations of distributive lattices and brouwerian logics. Casopis Pest. Math., 67:1--25, 1937.

....L = L; or with connectives added, merely conjunction (L; or with a full lattice structure (L; with or without negation operators of one sort or another, and possibly with additional operators. Stones s duality theorems for Boolean algebras and distributive lattices ( 30] 31] [32]) have initiated a well established tradition of representation and duality theorems in terms of compact, totally separated spaces (Stone spaces) Priestley ( 28, 29] in this tradition, proved a duality theorem for distributive lattices using partially ordered Stone spaces. In the case of ....

M. H. STONE (1937) "Topological Representation of Distributive Lattices and Brouwerian Logics", Casopsis pro Pestovani Matematiky a Fysiky 67, 1-25.

....system) and constructed from a boolean algebra a set of points using prime filters. Conversely, by using a topology on a set of points he was able to construct a boolean algebra. For certain topological spaces (later called Stone spaces) these constructions give an isomorphism. In a later paper [Sto37] he generalized this correspondence from Stone spaces to spectral spaces and from boolean algebra s to distributive lattices. Hofmann and Keimel [HK72] described this duality in a categorical framework. Even further, Isbell [Isb72a] gives an adjunction between the category of topological spaces ....

M.H. Stone. Topological Representation of Distributive Lattices and Brouwerian logics. Cas. Math. Fys., 67:1--25, 1937.

....constructed from a boolean algebra a set of points using completely prime filters. Conversely, by using a topology on a set of points he was able to construct a boolean algebra. For certain topological spaces (later called Stone spaces) these constructions give an isomorphism. In a later paper [Sto37] he generalized this correspondence from Stone spaces to spectral spaces and from boolean algebra s to distributive lattices. Hoffman and Keimel [HK72] described this duality in a categorical framework. Even further, Isbell [Isb72] gives an adjunction between the category of topological spaces ....

M.H. Stone. Topological Representation of Distributive Lattices and Brouwerian logics. Cas. Math. Fys., 67:1--25, 1937.

....2 fl; 62) R fffifl iff 8a; b; if b a 2 ff a 2 fi then b 2 fl: 63) We define ff v fi iff ff fi. It is easy to verify the tonicity conditions (30) 32) For the representation, we define the canonical isomorphism h(x) fff : x 2 ffg: 64) Relation Algebras 19 It is well known from Stone (1937) that this is a 1 1 map preserving and ; carrying them into intersection and union. It is also known, as a special case, from J onsson and Tarski (1951 52) that h preserves ffi; and Dunn (1991) has provided a generalization of the their work which contains as a special case that h preserves ....

M. Stone (1937), "Topological Representations of Distributive Lattices and Brouwerian Logics," Casopsis pro Pesto'van ' i Matematiky a Fysiky, 67, 1--25.

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M. H. Stone (1937) "Topological Representation of Distributive Lattices and Brouwerian Logics", Casopsis pro Pestovani Matematiky a Fysiky 67, 1-25.

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M. H. Stone (1937) "Topological Representation of Distributive Lattices and Brouwerian Logics", Casopsis pro Pestovani Matematiky a Fysiky 67, 1-25.

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Stone, M. H. (1937). Topological representations of distributive lattices and Brouwerian logics.

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Stone, M. (1937). Topological representations of distributive lattices and Brouwerian logics. Casopis Pest. Mat., 67:1--25.

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Stone, M. H., Topological representation of distributive lattices and

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Stone, M. H., Topological representations of distributive lattices and Browerian logics, Casopis. Pest. Mat. 67 (1937), 1-25.

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M. H. STONE (1937) "Topological Representation of Distributive Lattices and Brouwerian Logics", Casopsis pro Pestovani Matematiky a Fysiky 67, 1-25.

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M.H. Stone. Topological representation of distributive lattices and Brouwerian logics. Casopis Pro Potovan Mathematiky, 67:1--25, 1937.

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Stone, M., Topological representations of distributive lattices and Browerian logics. Casopis pest. Mat., Vol. 67, 1937, pages 1-25.

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M. Stone (1937), "Topological Representations of Distributive Lattices and Brouwerian Logics," Casopsis pro pesto'van ' i matematiky a fysiky, 67, 1--25.

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M. H. STONE (1937) "Topological Representation of Distributive Lattices and Brouwerian Logics", Casopsis pro Pestovani Matematiky a Fysiky 67, 1-25.

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M.H. Stone. Topological representation of distributive lattices and Brouwerian logics. Casopis Pro Potov'an'i Mathematiky, 67:1--25, 1937.

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