scholarly journals Quasi-N4-Lattices

2021 ◽  
Author(s):  
Umberto Rivieccio
Keyword(s):  
Paraconsistent Logic ◽  
General Setting ◽  
Extension Property ◽  
Double Negation ◽  
Structure Representation ◽  
Nelson Algebras ◽  
Nelson Logic ◽  
Double Negation Law ◽  
Logical Calculi ◽  
Twist Structure

Abstract Within the Nelson family, two mutually incomparable generalizations of Nelson constructive logic with strong negation have been proposed so far. The first and more well-known, Nelson paraconsistent logic , results from dropping the explosion axiom of Nelson logic; a more recent series of papers considers the logic (dubbed quasi-Nelson logic ) obtained by rejecting the double negation law, which is thus also weaker than intuitionistic logic. The algebraic counterparts of these logical calculi are the varieties of N4-lattices and quasi-Nelson algebras . In the present paper we propose the class of quasi- N4-lattices as a common generalization of both. We show that a number of key results, including the twist-structure representation of N4-lattices and quasi-Nelson algebras, can be uniformly established in this more general setting; our new representation employs twist-structures defined over Brouwerian algebras enriched with a nucleus operator. We further show that quasi-N4-lattices form a variety that is arithmetical, possesses a ternary as well as a quaternary deductive term, and enjoys EDPC and the strong congruence extension property.

On Bijective Correspondence between Fuzzy Reflexive Approximation Spaces and Fuzzy Transformation Systems

2020 ◽  
Vol 16 (02) ◽  
pp. 291-304
Author(s):  
Sutapa Mahato ◽  
S. P. Tiwari
Keyword(s):  
Lattice Structure ◽  
Approximation Space ◽  
Double Negation ◽  
Approximation Spaces ◽  
Bijective Correspondence ◽  
Fuzzy Approximation ◽  
Fuzzy Transformation ◽  
Double Negation Law ◽  
Transformation Systems ◽  
The Relationship

The objective of this paper is to establish the relationship between fuzzy approximation operators and fuzzy transformation systems. We show that for each upper fuzzy transformation system there exists a fuzzy reflexive approximation space and vice-versa. We further establish such relationship between lower fuzzy transformation systems and fuzzy reflexive approximation spaces under the condition that the underline lattice structure satisfies double negation law.

scholarly journals Residuation in non-associative MV-algebras

2018 ◽  
Vol 68 (6) ◽  
pp. 1313-1320
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Binary Operation ◽  
Residuated Lattice ◽  
Natural Question ◽  
Natural Conditions ◽  
Double Negation ◽  
Mv Algebra ◽  
Residuated Poset ◽  
Double Negation Law ◽  
Mv Algebras

Abstract It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In a previous paper the first author and J. Kühr introduced the concept of an NMV-algebra which is a non-associative modification of an MV-algebra. The natural question arises if an NMV-algebra can be converted into a residuated structure, too. Contrary to MV-algebras, NMV-algebras are not based on lattices but only on directed posets and the binary operation need not be associative and hence we cannot expect to obtain a residuated lattice but only an essentially weaker structure called a conditionally residuated poset. Considering several additional natural conditions we show that every NMV-algebra can be converted in such a structure. Also conversely, every such structure can be organized into an NMV-algebra. Further, we study an a bit more stronger version of an algebra where the binary operation is even monotone. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.

A minimum calculus for logic

1944 ◽  
Vol 9 (4) ◽  
pp. 89-94 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Truth Table ◽  
Additional Property ◽  
Basic Logic ◽  
Double Negation ◽  
Logical Calculus ◽  
Logical Calculi

A logical calculus will be presented which not only is a formulation of a “basic logic” in the sense of the writer's previous papers, but which has the additional property that no weaker calculus can be a formulation of a basic logic. A sort of minimum logical calculus is thus attained, which has nothing superfluous about it for achieving the purpose for which it is designed.In the case of some logical calculi the question can arise as to whether certain of the postulates are really logically valid and necessary. Sometimes a test is available, such as the truth-table test, enabling us to distinguish between logically valid sentences and others, but often no such test is available, especially where quantifiers are involved. Is or is not the axiom of infinity, for example, to be regarded as logically valid? Or is the principle of double negation really acceptable, even though it satisfies the truth-table test?

scholarly journals Residuation in orthomodular lattices

2017 ◽  
Vol 5 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Orthomodular Lattice ◽  
Residuated Lattice ◽  
Orthomodular Lattices ◽  
Double Negation ◽  
Converse Statement ◽  
One To One ◽  
Double Negation Law

Abstract We show that every idempotent weakly divisible residuated lattice satisfying the double negation law can be transformed into an orthomodular lattice. The converse holds if adjointness is replaced by conditional adjointness. Moreover, we show that every positive right residuated lattice satisfying the double negation law and two further simple identities can be converted into an orthomodular lattice. In this case, also the converse statement is true and the corresponence is nearly one-to-one.

scholarly journals Adjoint Operations in Twist-Products of Lattices

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 253
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Residuated Lattice ◽  
Sufficient Conditions ◽  
Residuated Lattices ◽  
Double Negation ◽  
Antitone Involution ◽  
Binary Operations ◽  
Double Negation Law ◽  
Commutative Residuated Lattice

Given an integral commutative residuated lattices L=(L,∨,∧), its full twist-product (L2,⊔,⊓) can be endowed with two binary operations ⊙ and ⇒ introduced formerly by M. Busaniche and R. Cignoli as well as by C. Tsinakis and A. M. Wille such that it becomes a commutative residuated lattice. For every a∈L we define a certain subset Pa(L) of L2. We characterize when Pa(L) is a sublattice of the full twist-product (L2,⊔,⊓). In this case Pa(L) together with some natural antitone involution ′ becomes a pseudo-Kleene lattice. If L is distributive then (Pa(L),⊔,⊓,′) becomes a Kleene lattice. We present sufficient conditions for Pa(L) being a subalgebra of (L2,⊔,⊓,⊙,⇒) and thus for ⊙ and ⇒ being a pair of adjoint operations on Pa(L). Finally, we introduce another pair ⊙ and ⇒ of adjoint operations on the full twist-product of a bounded commutative residuated lattice such that the resulting algebra is a bounded commutative residuated lattice satisfying the double negation law, and we investigate when Pa(L) is closed under these new operations.

scholarly journals When does a semiring become a residuated lattice?

2016 ◽  
Vol 09 (04) ◽  
pp. 1650088
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Residuated Lattice ◽  
Residuated Lattices ◽  
The Other ◽  
Natural Question ◽  
Double Negation ◽  
Complete Distributivity ◽  
Completely Distributive ◽  
Double Negation Law ◽  
The Given ◽  
Converse Assertion

It is an easy observation that every residuated lattice is in fact a semiring because multiplication distributes over join and the other axioms of a semiring are satisfied trivially. This semiring is commutative, idempotent and simple. The natural question arises if the converse assertion is also true. We show that the conversion is possible provided the given semiring is, moreover, completely distributive. We characterize semirings associated to complete residuated lattices satisfying the double negation law where the assumption of complete distributivity can be omitted. A similar result is obtained for idempotent residuated lattices.

Quasi-Nelson algebras and fragments

2021 ◽  
pp. 1-29
Author(s):  
Umberto Rivieccio ◽  
Ramon Jansana
Keyword(s):  
Residuated Lattices ◽  
Substructural Logic ◽  
Lambek Calculus ◽  
Nelson Algebras ◽  
Algebraic Language ◽  
Nelson Logic ◽  
Algebraic Counterpart

Abstract The variety of quasi-Nelson algebras (QNAs) has been recently introduced and characterised in several equivalent ways: among others, as (1) the class of bounded commutative integral (but non-necessarily involutive) residuated lattices satisfying the Nelson identity, as well as (2) the class of (0, 1)-congruence orderable commutative integral residuated lattices. Logically, QNAs are the algebraic counterpart of quasi-Nelson logic, which is the (algebraisable) extension of the substructural logic ℱℒ ew (Full Lambek calculus with Exchange and Weakening) by the Nelson axiom. In the present paper, we collect virtually all the results that are currently known on QNAs, including solutions to certain questions left open in earlier publications. Furthermore, we extend our study to some subreducts of QNAs, that is, classes of algebras corresponding to fragments of the algebraic language obtained by eliding either the implication or the lattice operations.

scholarly journals Negation and Implication in Quasi-Nelson Logic

2021 ◽  
Vol 27 (1) ◽  
pp. 107-123
Author(s):  
Thiago Nascimento ◽  
Umberto Rivieccio
Keyword(s):  
Intuitionistic Logic ◽  
Constructive Logic ◽  
Strong Negation ◽  
Completeness Result ◽  
Nelson Algebras ◽  
Nelson Logic

Quasi-Nelson logic is a recently-introduced generalization of Nelson’s constructive logic with strong negation to a non-involutive setting. In the present paper we axiomatize the negation-implication fragment of quasi-Nelson logic (QNI-logic), which constitutes in a sense the algebraizable core of quasi-Nelson logic. We introduce a finite Hilbert-style calculus for QNI-logic, showing completeness and algebraizability with respect to the variety of QNI-algebras. Members of the latter class, also introduced and investigated in a recent paper, are precisely the negation-implication subreducts of quasi-Nelson algebras. Relying on our completeness result, we also show how the negation-implication fragments of intuitionistic logic and Nelson’s constructive logic may both be obtained as schematic extensions of QNI-logic.

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