Residuated Structures and Orthomodular Lattices

Studia Logica ◽

10.1007/s11225-021-09946-1 ◽

2021 ◽

Author(s):

D. Fazio ◽

A. Ledda ◽

F. Paoli

Keyword(s):

Algebraic Logic ◽

Residuated Lattices ◽

Quantum Structures ◽

Heyting Algebras ◽

Quantum Algebras ◽

Orthomodular Lattices ◽

Lattice Of Subvarieties ◽

Residuated Structures ◽

Common Framework ◽

Mv Algebras

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.

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Perfect residuated lattice ordered monoids

Mathematica Slovaca ◽

10.2478/s12175-010-0050-6 ◽

2010 ◽

Vol 60 (6) ◽

Cited By ~ 1

Author(s):

Jiří Rachůnek ◽

Dana Šalounová

Keyword(s):

Probability Measure ◽

Residuated Lattice ◽

Residuated Lattices ◽

Heyting Algebras ◽

Effect Algebras ◽

Quantum Logics ◽

Mv Algebras ◽

Intuitionistic Logics

AbstractBounded Rℓ-monoids form a large subclass of the class of residuated lattices which contains certain of algebras of fuzzy and intuitionistic logics, such as GMV-algebras (= pseudo-MV-algebras), pseudo-BL-algebras and Heyting algebras. Moreover, GMV-algebras and pseudo-BL-algebras can be recognized as special kinds of pseudo-MV-effect algebras and pseudo-weak MV-effect algebras, i.e., as algebras of some quantum logics. In the paper, bipartite, local and perfect Rℓ-monoids are investigated and it is shown that every good perfect Rℓ-monoid has a state (= an analogue of probability measure).

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Basic algebras, logics, trends and applications

Asian-European Journal of Mathematics ◽

10.1142/s1793557115500400 ◽

2015 ◽

Vol 08 (03) ◽

pp. 1550040 ◽

Cited By ~ 5

Author(s):

Ivan Chajda

Keyword(s):

Quantum Mechanics ◽

20Th Century ◽

19Th Century ◽

Classical Logic ◽

Boolean Algebras ◽

Heyting Algebras ◽

Fuzzy Logics ◽

Orthomodular Lattices ◽

Mv Algebras ◽

Intuitionistic Logics

The classical logic was axiomatized algebraically by means of Boolean algebras in 19th century by George Boole. Similar attempts went on 20th century for algebraic axiomatization of non-classical logics, e.g. intuitionistic logics (Brouwer and Heyting algebras), many-valued logics (Łukasiewicz, Chang’s MV-algebras, Post algebras), the logic of quantum mechanics (orthomodular lattices and posets) and fuzzy logics (residuated lattices). In this paper, we are focused in a common generalization of MV-algebras and orthomodular lattices. The resulting algebras, called basic algebras, have surprisingly strong and interesting properties and they can be investigated in their own. The aim of the paper is to get an overview of results reached during the last decade.

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Splittings in varieties of logic

International Journal of Algebra and Computation ◽

10.1142/s021819672150034x ◽

2021 ◽

pp. 1-48

Author(s):

Brian A. Davey ◽

Tomasz Kowalski ◽

Christopher J. Taylor

Keyword(s):

Residuated Lattices ◽

Heyting Algebras ◽

General Lemma ◽

Negative Results ◽

Splitting Algebras

We study splittings or lack of them, in lattices of subvarieties of some logic-related varieties. We present a general lemma, the non-splitting lemma, which when combined with some variety-specific constructions, yields each of our negative results: the variety of commutative integral residuated lattices contains no splitting algebras, and in the varieties of double Heyting algebras, dually pseudocomplemented Heyting algebras and regular double [Formula: see text]-algebras the only splitting algebras are the two-element and three-element chains.

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L-algebras and topology

Journal of Algebra and Its Applications ◽

10.1142/s0219498823500342 ◽

2021 ◽

Author(s):

Wolfgang Rump

Keyword(s):

Mapping Class Groups ◽

Unitary Groups ◽

Mapping Class ◽

Heyting Algebras ◽

Baxter Equation ◽

Prime Spectrum ◽

Class Groups ◽

Orthomodular Lattices ◽

And Topology ◽

Torsion Free

[Formula: see text]-algebras are based on an equation which is fundamental in the construction of various torsion-free groups, including spherical Artin groups, Riesz groups, certain mapping class groups, para-unitary groups, and structure groups of set-theoretic solutions to the Yang–Baxter equation. A topological study of [Formula: see text]-algebras is initiated. A prime spectrum is associated to certain (possibly all) [Formula: see text]-algebras, including three classes of [Formula: see text]-algebras where the ideals are determined in a more explicite fashion. Known results on orthomodular lattices, Heyting algebras, or quantales are extended and revisited from an [Formula: see text]-algebraic perspective.

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Left residuated lattices induced by lattices with a unary operation

Soft Computing ◽

10.1007/s00500-019-04461-x ◽

2019 ◽

Vol 24 (2) ◽

pp. 723-729

Author(s):

Ivan Chajda ◽

Helmut Länger

Keyword(s):

Orthomodular Lattice ◽

Residuated Lattice ◽

Unary Operation ◽

Residuated Lattices ◽

Boolean Algebras ◽

Natural Question ◽

Similar Technique ◽

Orthomodular Lattices

Abstract In a previous paper, the authors defined two binary term operations in orthomodular lattices such that an orthomodular lattice can be organized by means of them into a left residuated lattice. It is a natural question if these operations serve in this way also for more general lattices than the orthomodular ones. In our present paper, we involve two conditions formulated as simple identities in two variables under which this is really the case. Hence, we obtain a variety of lattices with a unary operation which contains exactly those lattices with a unary operation which can be converted into a left residuated lattice by use of the above mentioned operations. It turns out that every lattice in this variety is in fact a bounded one and the unary operation is a complementation. Finally, we use a similar technique by using simpler terms and identities motivated by Boolean algebras.

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Partial residuated structures and quantum structures

Soft Computing ◽

10.1007/s00500-008-0283-2 ◽

2008 ◽

Vol 12 (12) ◽

pp. 1219-1227 ◽

Cited By ~ 3

Author(s):

Xiangnan Zhou ◽

Qingguo Li

Keyword(s):

Quantum Structures ◽

Residuated Structures

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Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras

Bulletin of the Section of Logic ◽

10.18778/0138-0680.45.3.4.08 ◽

2016 ◽

Vol 45 (3/4) ◽

Author(s):

Wojciech Dzik ◽

Sándor Radeleczki

Keyword(s):

Residuated Lattices ◽

Heyting Algebras ◽

Direct Products ◽

Unification Type

We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider examples of frontal Heyting algebras, in particular Heyting algebras with the successor, γ and G operations as well as expansions of some commutative integral residuated lattices with successor operations.

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Quasi-subtractive varieties

Journal of Symbolic Logic ◽

10.2178/jsl/1318338848 ◽

2011 ◽

Vol 76 (4) ◽

pp. 1261-1286 ◽

Cited By ~ 9

Author(s):

Tomasz Kowalski ◽

Francesco Paoli ◽

Matthew Spinks

Keyword(s):

Algebraic Logic ◽

Substructural Logics ◽

Boolean Algebras ◽

Normal Subgroups ◽

Regular Variety ◽

Quantum Logics ◽

Algebraic Investigation ◽

Lattice Of Congruences ◽

Isomorphism Theorems ◽

Mv Algebras

AbstractVarieties like groups, rings, or Boolean algebras have the property that, in any of their members, the lattice of congruences is isomorphic to a lattice of more manageable objects, for example normal subgroups of groups, two-sided ideals of rings, filters (or ideals) of Boolean algebras. Abstract algebraic logic can explain these phenomena at a rather satisfactory level of generality: in every member A of a τ-regular variety the lattice of congruences of A is isomorphic to the lattice of deductive filters on A of the τ-assertional logic of . Moreover, if has a constant 1 in its type and is 1-subtractive, the deductive filters on A ∈ of the 1-assertional logic of coincide with the -ideals of A in the sense of Gumm and Ursini, for which we have a manageable concept of ideal generation.However, there are isomorphism theorems, for example, in the theories of residuated lattices, pseudointerior algebras and quasi-MV algebras that cannot be subsumed by these general results. The aim of the present paper is to appropriately generalise the concepts of subtractivity and τ-regularity in such a way as to shed some light on the deep reason behind such theorems. The tools and concepts we develop hereby provide a common umbrella for the algebraic investigation of several families of logics, including substructural logics, modal logics, quantum logics, and logics of constructive mathematics.

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