Basics of Algebraic Logic

AbstractThe variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., $$\ell $$ ℓ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated $$\ell $$ ℓ -groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated $$\ell $$ ℓ -groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated $$\ell $$ ℓ -groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.

Aubert Daigneault. Introduction. Studies in algebraic logic, edited by Aubert Daigneault, Studies in mathematics, vol. 9, The Mathematical Association of America, [Washington, D.C.], 1974, pp. 1–5. - William Craig. Unification and abstraction in algebraic logic. Studies in algebraic logic, edited by Aubert Daigneault, Studies in mathematics, vol. 9, The Mathematical Association of America, [Washington, D.C.], 1974, pp. 6–57. - J. Donald Monk. Connections between combinatorial theory and algebraic logic. Studies in algebraic logic, edited by Aubert Daigneault, Studies in mathematics, vol. 9, The Mathematical Association of America, [Washington, D.C.], 1974, pp. 58–91. - Helena Rasiowa. Post algebras as a semantic foundation of m-valued logics. Studies in algebraic logic, edited by Aubert Daigneault, Studies in mathematics, vol. 9, The Mathematical Association of America, [Washington, D.C.], 1974, pp. 92–142. - Gonzalo E. Reyes. From sheaves to logic. Studies in algebraic logic, edited by Aubert Daigneault, Studies in mathematics, vol. 9, The Mathematical Association of America, [Washington, D.C.], 1974, pp. 143–204.

Journal of Symbolic Logic ◽

10.2307/2271959 ◽

1978 ◽

Vol 43 (1) ◽

pp. 145-147

Author(s):

Anne Preller

Keyword(s):

Algebraic Logic ◽

Combinatorial Theory ◽

Mathematical Association

An abstract algebraic logic approach to tetravalent modal logics

Journal of Symbolic Logic ◽

10.2307/2586552 ◽

2000 ◽

Vol 65 (2) ◽

pp. 481-518 ◽

Cited By ~ 15

Author(s):

Josep Maria Font ◽

Miquel Rius

Keyword(s):

Algebraic Logic ◽

Modal Logics ◽

The Other ◽

Abstract Algebraic Logic ◽

Algebraic Semantics ◽

Modal Algebras ◽

Logic Approach ◽

Abstract Logics ◽

Joint Study

AbstractThis paper contains a joint study of two sentential logics that combine a many-valued character, namely tetravalence, with a modal character; one of them is normal and the other one quasinormal. The method is to study their algebraic counterparts and their abstract models with the tools of Abstract Algebraic Logic, and particularly with those of Brown and Suszko's theory of abstract logics as recently developed by Font and Jansana in their “A General Algebraic Semantics for Sentential Logics”. The logics studied here arise from the algebraic and lattice-theoretical properties we review of Tetravalent Modal Algebras, a class of algebras studied mainly by Loureiro, and also by Figallo. Landini and Ziliani, at the suggestion of the late Antonio Monteiro.

Categorical Abstract Algebraic Logic: Equivalential π-Institutions

The Australasian Journal of Logic ◽

10.26686/ajl.v6i0.1790 ◽

2008 ◽

Vol 6 ◽

Cited By ~ 2

Author(s):

George Voutsadakis

Keyword(s):

Matrix Theory ◽

Algebraic Logic ◽

Characterization Theorem ◽

Abstract Algebraic Logic ◽

Transfer Theorem ◽

Deductive Systems

The theory of equivalential deductive systems, as introduced by Prucnal and Wrónski and further developed by Czelakowski, is abstracted to cover the case of logical systems formalized as π-institutions. More precisely, the notion of an N-equivalence system for a given π-institution is introduced. A characterization theorem for N-equivalence systems, previously proven for N-parameterized equivalence systems, is revisited and a “transfer theorem” for N-equivalence systems is proven. For a π-institution I having an N-equivalence system, the maximum such system is singled out and, then, an analog of Herrmann’s Test, characterizing those N-protoalgebraic π-institutions having an N-equivalence system, is formulated. Finally, some of the rudiments of matrix theory are revisited in the context of π-institutions, as they relate to the existence of N-equivalence systems.

Categorical Abstract Algebraic Logic: More on Protoalgebraicity

Notre Dame Journal of Formal Logic ◽

10.1305/ndjfl/1168352663 ◽

2006 ◽

Vol 47 (4) ◽

pp. 487-514 ◽

Cited By ~ 4

Author(s):

George Voutsadakis

Keyword(s):

Algebraic Logic ◽

Abstract Algebraic Logic

Assertional logics, truth-equational logics, and the hierarchies of abstract algebraic logic

Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science - Outstanding Contributions to Logic ◽

10.1007/978-3-319-74772-9_2 ◽

2018 ◽

pp. 53-79 ◽

Cited By ~ 1

Author(s):

Hugo Albuquerque ◽

Josep Maria Font ◽

Ramon Jansana ◽

Tommaso Moraschini

Keyword(s):

Algebraic Logic ◽

Abstract Algebraic Logic

ALGORITHMS FOR CONTROLLING LARGE NAVIGATION AND CONTROL INFORMATION BRIDGE COMPLEXES USING THE FUNCTIONS OF THE ALGEBRAIC LOGIC OF PROGRAMMABLE BLOCKS

Transport Business of Russia ◽

10.52375/20728689_2021_2_128 ◽

2021 ◽

pp. 128-133

Author(s):

A. Popov ◽

A. Troeglazov

Keyword(s):

Algebraic Logic ◽

Control Information ◽

Navigation And Control ◽

And Control

The Categorial Approach to Algebraic Logic

Universal Algebra, Algebraic Logic, and Databases ◽

10.1007/978-94-011-0820-1_12 ◽

1994 ◽

pp. 247-274

Author(s):

B. Plotkin

Keyword(s):

Algebraic Logic

Automating Algebraic Proofs in Algebraic Logic

Fundamenta Informaticae ◽

10.3233/fi-1996-281208 ◽

1996 ◽

Vol 28 (1,2) ◽

pp. 129-140

Author(s):

Jieh Hsiang ◽

Anita Wasilewska

Keyword(s):

Algebraic Logic ◽

Algebraic Proofs

Varieties of distributive lattices with unary operations I

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s144678870000063x ◽

1997 ◽

Vol 63 (2) ◽

pp. 165-207 ◽

Cited By ~ 7

Author(s):

H. A. Priestley

Keyword(s):

Duality Theory ◽

Algebraic Logic ◽

Distributive Lattices ◽

Ordered Semigroup ◽

Finitely Generated ◽

Left Multiplication ◽

Unified Study

AbstractA unified study is undertaken of finitely generated varieties HSP () of distributive lattices with unary operations, extending work of Cornish. The generating algebra () is assusmed to be of the form (P; ∧, ∨, 0, 1, {fμ}), where each fμ is an endomorphism or dual endomorphism of (P; ∧, ∨, 0, 1), and the Priestly dual of this lattice is an ordered semigroup N whose elements act by left multiplication to give the maps dual to the operations fμ. Duality theory is fully developed within this framework, into which fit many varieties arising in algebraic logic. Conditions on N are given for the natural and Priestley dualities for HSP () to be essentially the same, so that, inter alia, coproducts in HSP () are enriched D-coproducts.