Filters of strong Sheffer stroke non-associative MV-algebras

Abstract In this paper, at first we study strong Sheffer stroke NMV-algebra. For getting more results and some classification, the notions of filters and subalgebras are introduced and studied. Finally, by a congruence relation, we construct a quotient strong Sheffer stroke NMV-algebra and isomorphism theorems are proved.

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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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Sheffer Stroke Nelson Algebras

10.21203/rs.3.rs-712955/v1 ◽

2021 ◽

Author(s):

Tahsin Oner ◽

Tugce Katican ◽

Arsham Borumand Saeid

Keyword(s):

Modular Lattice ◽

Congruence Relation ◽

Algebraic Structures ◽

Sheffer Stroke ◽

Nelson Algebra ◽

The Family ◽

Nelson Algebras

Abstract In this study, Sheffer stroke Nelson algebras (briefly, s-Nelson algebras), (ultra) ideals, quasi-subalgebras and quotient sets on these algebraic structures are introduced. The relationships between s-Nelson and Nelson algebras are analyzed. Also, it is shown that a s-Nelson algbera is a bounded distributive modular lattice, and the family of all ideals forms a complete distributive modular lattice. A congruence relation on s-Nelson algebra is determined by its ideal and quotient s-Nelson algebras are constructed by this congruence relation. Finally, it is indicated that a quotient s-Nelson algebra defined by the ultra ideal is totally ordered and that the cardinality of the quotient is less than or equals to 2.

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Ideals in residuated lattices

Carpathian Journal of Mathematics ◽

10.37193/cjm.2021.01.06 ◽

2021 ◽

Vol 37 (1) ◽

pp. 53-63

Author(s):

DUMITRU BUŞNEG ◽

DANA PICIU ◽

ANCA-MARIA DINA

Keyword(s):

Fuzzy Sets ◽

Binary Relation ◽

Congruence Relation ◽

Natural Generalization ◽

Residuated Lattices ◽

Algebraic Analysis ◽

Deductive Systems ◽

Mv Algebra ◽

The Relationship ◽

Mv Algebras

"The notion of ideal in residuated lattices is introduced in [Kengne, P. C., Koguep, B. B., Akume, D. and Lele, C., L-fuzzy ideals of residuated lattices, Discuss. Math. Gen. Algebra Appl., 39 (2019), No. 2, 181–201] and [Liu, Y., Qin, Y., Qin, X. and Xu, Y., Ideals and fuzzy ideals in residuated lattices, Int. J. Math. Learn & Cyber., 8 (2017), 239–253] as a natural generalization of that of ideal in MV algebras (see [Cignoli, R., D’Ottaviano, I. M. L. and Mundici, D., Algebraic Foundations of many-valued Reasoning, Trends in Logic-Studia Logica Library 7, Dordrecht: Kluwer Academic Publishers, 2000] and [Chang, C. C., Algebraic analysis of many-valued logic, Trans. Amer. Math. Soc., 88 (1958), 467–490]). If A is an MV algebra and I is an ideal on A then the binary relation x ∼I y iff x^{*}Ꙩ y; x Ꙩy^{*} ∈ I , for x; y ∈ A; is a congruence relation on A. In this paper we find classes of residuated lattices for which the relation ∼ I (defined for MV algebras) is a congruence relation and we give new characterizations for i-ideals and prime i-ideals in residuated lattices. As a generalization of the case of BL algebras (see [Lele, C. and Nganou, J. B., MV-algebras derived from ideals in BL-algebras, Fuzzy Sets and Systems, 218 (2013), 103–113]), we investigate the relationship between i-ideals and deductive systems in residuated lattices."

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Semi-simple and complete MV-algebras

Algebra Universalis ◽

10.1007/bf01190751 ◽

1992 ◽

Vol 29 (1) ◽

pp. 1-9 ◽

Cited By ~ 37

Author(s):

L. P. Belluce

Keyword(s):

Mv Algebras

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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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Priestley duality for MV-algebras and beyond

Forum Mathematicum ◽

10.1515/forum-2020-0115 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Wesley Fussner ◽

Mai Gehrke ◽

Samuel J. van Gool ◽

Vincenzo Marra

Keyword(s):

Large Class ◽

Order Conditions ◽

Enriched Environment ◽

Priestley Duality ◽

Distributive Lattices ◽

First Order ◽

Dual Spaces ◽

Binary Operations ◽

New Perspective ◽

Mv Algebras

Abstract We provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators. Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order conditions on dual spaces. In particular, we specialize our general results to the variety of MV-algebras, obtaining a duality for these in which the equations axiomatizing MV-algebras are dualized as first-order conditions.

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On fundamental isomorphism theorems in soft subgroups

Soft Computing ◽

10.1007/s00500-020-05074-5 ◽

2020 ◽

Vol 24 (16) ◽

pp. 11841-11851

Author(s):

N. Çağman ◽

R. Barzegar ◽

S. B. Hosseini

Keyword(s):

Isomorphism Theorems

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Congruence relations on almost distributive lattices-II

Asian-European Journal of Mathematics ◽

10.1142/s1793557121500054 ◽

2019 ◽

pp. 2150005

Author(s):

Gezahagne Mulat Addis

Keyword(s):

Distributive Lattice ◽

Congruence Relation ◽

Congruence Class ◽

Distributive Lattices ◽

Ideal Formula ◽

Congruence Relations

For a given ideal [Formula: see text] of an almost distributive lattice [Formula: see text], we study the smallest and the largest congruence relation on [Formula: see text] having [Formula: see text] as a congruence class.

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