Sur les algèbres de Hilbert
About: This article is published in Journal of Symbolic Logic.The article was published on 1970-03-01. It has received 105 citations till now.
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02 Oct 1996TL;DR: This monograph develops a very general approach to the algebra- ization of sententiallogics, to show its results on a number of particular logics, and to relate it to other existing approaches, namely to those based on logical matrices and the equational consequence developed by Blok, Czelakowski, Pigozzi and others.
Abstract: The purpose of this monograph is to develop a very general approach to the algebra- ization of sententiallogics, to show its results on a number of particular logics, and to relate it to other existing approaches, namely to those based on logical matrices and the equational consequence developed by Blok, Czelakowski, Pigozzi and others. The main distinctive feature of our approachlies in the mathematical objects used as models of a sententiallogic: We use abstract logics, while the dassical approaches use logical matrices. Using models with more structure allows us to reflect in them the metalogical properties of the sentential logic. Since an abstract logic can be viewed as a "bundle" or family of matrices, one might think that the new models are essentially equivalent to the old ones; but we believe, after an overall appreciation of the work done in this area, that it is precisely the treatment of an abstract logic as a single object that gives rise to a useful -and beautiful- mathematical theory, able to explain the connections, not only at the logical Ievel but at the metalogical Ievel, between a sentential logic and the particular dass of models we associate with it, namely the dass of its full models. Traditionally logical matrices have been regarded as the most suitable notion of model in the algebraic studies of sentential logics; and indeed this notion gives sev- eral completeness theorems and has generated an interesting mathematical theory.
201 citations
TL;DR: The notion of apseudo-interior algebras was introduced in this paper, which is a hybrid of a topological interior algebra and a residuated partially ordered monoid.
Abstract: The notion of apseudo-interior algebra is introduced; it is a hybrid of a (topological) interior algebra and a residuated partially ordered monoid. The elementary arithmetic of pseudo-interior algebras is developed leading to a simple equational axiomatization. A notion ofopen filter analogous to the open filters of interior algebras is investigated. Pseudo-interior algebras represent, in algebraic form, the logic inherent in varieties with acommutative, regular ternary deductive (TD) term p(x, y, z), which is defined by the conditions: (1)p(x,y,z) ≡ z (modΘ(x, y)); (2) for fixed elementsa, b of an algebra A, {p(a, b, z):z ∈ A} is a transversal of the set of equivalence classes of Θ(a, b); (3)p(a, b, z) andp(a′,b′,z) define the same transversal wheneverΘ(a,b)=Θ(a′,b′); (4)Θ(p(x, y, 1), 1)= Θ(x, y) for some constant term 1. The TD term generalizes the (affine) ternary discriminator. Varieties with a commutative, regular TD term include most of the varieties of traditional algebraic logic as well as all double-pointed affine discriminator varieties andn-potent hoops (residuated commutative po-monoids in which the partial ordering is inverse divisibility). The main theorem:A variety has a commutative, regular TD term iff it is termwise definitionally equivalent to a pseudo-interior algebra with additional operations that are compatible with the open filters in a natural way.
157 citations
TL;DR: In this paper, a new and useful criterion for a variety to be locally finite is presented, and many examples are given to justify the effectiveness of the criterion, and the criterion is shown to be useful for many problems.
Abstract: In this paper we present a new and useful criterion for a variety to be locally finite. Many examples are given to justify the effectiveness of the criterion.
40 citations
Cites background from "Sur les algèbres de Hilbert"
...The investigation of implicational semilattices can be found in Diego [11], Köhler [18] and Rasiowa [29]....
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...implicational semilattices can be found in Diego [11], Köhler [18] and Rasiowa [29]....
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TL;DR: A new simplified proof is obtained of Zakharyaschev’s theorem that each intermediate logic can be axiomatized by canonical formulas by establishing a generalization of Esakia duality.
Abstract: We introduce partial Esakia morphisms, well partial Esakia morphisms, and strong partial Esakia morphisms between Esakia spaces and show that they provide the dual description of (∧,→) homomorphisms, (∧,→, 0) homomorphisms, and (∧,→,∨) homomorphisms between Heyting algebras, thus establishing a generalization of Esakia duality. This yields an algebraic characterization of Zakharyaschev’s subreductions, cofinal subreductions, dense subreductions, and the closed domain condition. As a consequence, we obtain a new simplified proof (which is algebraic in nature) of Zakharyaschev’s theorem that each intermediate logic can be axiomatized by canonical formulas. §
36 citations
Cites background or methods from "Sur les algèbres de Hilbert"
...Another useful tool of algebraic nature is Diego’s theorem [11] that the variety of implicative meet semilattices is locally finite....
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...On the other hand, as follows from Diego (1966), the variety of implicative meet-semilattices is locally finite....
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...Another useful tool of algebraic nature is Diego’s (1966) theorem that the variety of implicative meet-semilattices is locally finite....
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...On the other hand, as follows from Diego [11], the variety of implicative meet-semilattices is locally finite....
[...]
35 citations