Involutive right-residuated l-groupoids

Chajda, Ivan; Radeleczki, Sándor

doi:10.1007/s00500-015-1765-7

Cited by 3 publications

(7 citation statements)

References 16 publications

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“…Analogues of Lemma 35 have been provided for several subvarieties of PLRG. In particular, Chajda and Radelecki [23] come very close to the generality attained in this lemma, since the only additional assumption they make on their algebras are 0-boundedness and integrality.…”

Section: Lemma 35 Plrg Is An Arithmetical and 1-ideal-determined Varietymentioning

confidence: 61%

“…This subvariety, here referred to as PCRG, has been the object of extensive investigations from both the algebraic and the proof-theoretical perspective (see e.g. [23,29,34]). This applies, in particular, to its most prominent subvarieties, such as the varieties PCRL of pointed commutative residuated lattices, MV of MV-algebras and H of Heyting algebras.…”

Section: Some Notable Subvarietiesmentioning

confidence: 99%

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Residuated Structures and Orthomodular Lattices

2021

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The 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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Section: Lemma 35 Plrg Is An Arithmetical and 1-ideal-determined Varietymentioning

confidence: 61%

Section: Some Notable Subvarietiesmentioning

confidence: 99%

Residuated Structures and Orthomodular Lattices

2021

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“…By a different approach, which relies on the order structure of commutative wBCK-algebras, this result is obtained in subsection 6.1 of [28]. A similar dual construction in a context not related to wBCK*-algebras is presented also in [13], [17], [21] and [22]. The algebra N 2 5 just described shows that a wBCK ∧ -algebra is not necessarily commutative.…”

Section: Commutative Wbck-algebrasmentioning

confidence: 84%

“…We also disclose in Sections 3-5 that a number of algebras with implication known in the literature (see [3,8,9,10,13,14,17,18,19,20,21,22,34]) are, in fact, commutative wBCK*-algebras of that or other type.…”

Section: Introductionmentioning

confidence: 94%

“…Another particular case, With some inessential distinctions, a dual set of axioms appears in other connection in Section 6.4 of [17]. Equivalent versions of it are used in [13], [21] and [22] to define respectively the classes of Abbot groupoids, implication basic algebras and so called strong I-algebras (cf. also [11,Lemma 1]).…”

Section: Commutative Wbck-algebrasmentioning

confidence: 99%

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On Some Classes of Commutative Weak BCK-Algebras

Cı̄rulis

2014

Stud Logica

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The class of weak BCK-algebras can be obtained by replacing (in the standard axiom set by K. Iseki and S. Tanaka) the first BCK axiomIt is known that every weak BCK-algebra is completely determined by the structure of its initial segments. We review several natural classes of commutative weak BCKalgebras, prove that they are equationally definable, and show that the order duals of a multitude of algebras with implication known in the literature in connection with various quantum logics are, in fact, commutative weak BCKalgebras belonging to that or other of these classes. We also characterize initial segments of algebras in each of the classes as lattices equipped with a suitable kind of complementation. In particular, commutative weak BCK-algebras are just those meet semilattices with the least element in which all initial segments are non-distributive De Morgan lattices.2010 Mathematics Subject Classification. Primary 06F35; Secondary 03G25, 06B75, 06C15.

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A Substructural Gentzen Calculus for Orthomodular Quantum Logic

Fazio¹,

Ledda²,

Paoli³

et al. 2022

The Review of Symbolic Logic

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We introduce a sequent system which is Gentzen algebraisable with orthomodular lattices as equivalent algebraic semantics, and therefore can be viewed as a calculus for orthomodular quantum logic. Its sequents are pairs of non-associative structures, formed via a structural connective whose algebraic interpretation is the Sasaki product on the left-hand side and its De Morgan dual on the right-hand side. It is a substructural calculus, because some of the standard structural sequent rules are restricted—by lifting all such restrictions, one recovers a calculus for classical logic.

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Involutive right-residuated l-groupoids

Cited by 3 publications

References 16 publications

Residuated Structures and Orthomodular Lattices

Residuated Structures and Orthomodular Lattices

On Some Classes of Commutative Weak BCK-Algebras

A Substructural Gentzen Calculus for Orthomodular Quantum Logic

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