Operations and structures derived from non-associative MV-algebras

Abstract

The so-called non-associative MV-algebras were introduced recently by the first author and J. Kühr in order to have an appropriate tool for certain logics used in expert systems where associativity of the binary operation is excluded, see, e.g., Botur and Halaš (Arch Math Log 48:243–255, 2009 ). Since implication is an important logical connective in practically every propositional logic, in the present paper we investigate the implication reducts of non-associative MV-algebras. We also determine their structures based on the underlying posets. The natural question when a poset with the greatest element equipped with sectional switching involutions can be organized into an implication NMV-algebra is solved. Moreover, congruence properties of the variety of implication NMV-algebras with, respectively, without zero are investigated. Analogously to classical propositional logic, we introduce a certain kind of Sheffer operation and we obtain a one-to-one correspondence between NMV-algebras and certain algebras built up by a Sheffer-like operation together with a unary operation.

Neutrosophic N-structures on Sheffer stroke BE-algebras

“…Conversely, Let S N be a neutrosophic N −structure on S satisfying the condition (5). Then it is obtained from Lemma 1 (i) and the condition (5…”

“…Sheffer stroke which is one of the two operators that can be used by itself, without any other logical operators, was originally introduced by H. M. Sheffer to build a logical formal system [22]. Since it provides new, basic and easily applicable axiom systems for many algebraic structures owing to its commutative property, this operation has many applications in algebraic structures such as orthoimplication algebras [1], ortholattices [4], Boolean algebras [11], strong Sheffer stroke non-associative MV-algebras [5], filters [14] and neutrosophic N -structures [17], Sheffer Stroke Hilbert Algebras [12], fuzzy filters [13] and neutrosophic N -structures [18], (fuzzy) filters of Sheffer stroke BL-algebras [15] and Sheffer stroke UP-algebras [16]. On the other hand, H S. Kim and Y. H. Kim introduced BE-algebras as a generalization of a dual BCK-algebra and defined filters and upper sets on this algebra [10].…”

Sheffer Stroke Nelson Algebras

“…Therefore, there exist its some applications in computer science, technology and industry. Besides, this binary operation has been used in algebraic structures such as orthoimplication algebras [1], ortholattices [2], strong Sheffer stroke non-associative MV-algebras [3], filters [12] and neutrosophic N -structures [17], Sheffer stroke Hilbert algebras [13], fuzzy filters [14] and neutrosophic N -structures [18], (fuzzy) filters and neutrosophic N-structures of Sheffer stroke BL-algebras ( [15], [8]), Sheffer stroke UP-algebras [16], Sheffer stroke BE-algebras [7], Sheffer stroke BG-algebras [24] and their fuzzy implivative ideals [20] and Sheffer stroke BCK-algebras [21].…”

Regular CA-Groupoids and Cyclic Associative Neutrosophic Extended Triplet Groupoids (CA-NET-Groupoids) with Green Relations

“…Moreover, Zhan and Tan [10] introduced the notion of left weakly Novikov algebra. In many fields (such as non-associative rings and non-associative algebras [11][12][13][14]), image processing [15], and networks [16]), non-associativity has essential research significance. Since cyclic associative law is widely used in algebraic systems, we have been focusing on the basic algebraic structure of cyclic associative groupoids (CA-groupoids) and other relevant algebraic structures (see [17,18]).…”