Neutrosophic Filters in Pseudo-Bci Algebras

2019

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The quasi-cancellativity of Abel Grassmann's groupoids (AG-groupoids) are discussed and two conjectures are partially solved. First, the following conjecture is proved to be true: every AG-3-band is quasi-cancellative. Moreover, a new notion of AG-(4,1)-band is proposed, and it is also proved that every AG-(4,1)-band is quasi-cancellative. Second, the notions of left (right) quasi-cancellative AG-groupoids and power-cancellative AG-groupoids are proposed, and the following results are obtained: for an AG*-groupoid or AG**-groupoid, it is left quasi-cancellative if and only if it is right quasi-cancellative; for a power-cancellative and locally power-associative AG-groupoid, it is left quasi-cancellative if and only if it is right quasi-cancellative. Finally, a general result is proved, that for any AG-groupoid, if it is left quasi-cancellative then it is right quasi-cancellative. Conjecture 1. Conditions (i) and (ii) above are equivalent for AG-groupoids. Conjecture 2. Every AG-3-band is quasi-cancellative.

Section: Discussionmentioning

confidence: 99%

On Two Conjectures of Abel Grassmann’s Groupoids

2019

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“…Ma and the decomposition theorem of PWC-NET-semihypergroup was proved in Reference [23]. For the study of some related algebraic systems, please refer to Reference [24][25][26].…”

Section: Introductionmentioning

confidence: 99%

On Neutrosophic Extended Triplet LA-hypergroups and Strong Pure LA-semihypergroups

Smarandache

2020

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We introduce the notions of neutrosophic extended triplet LA-semihypergroup, neutrosophic extended triplet LA-hypergroup, which can reflect some symmetry of hyperoperation and discuss the relationships among them and regular LA-semihypergroups, LA-hypergroups, regular LA-hypergroups. In particular, we introduce the notion of strong pure neutrosophic extended triplet LA-semihypergroup, get some special properties of it and prove the construction theorem about it under the condition of asymmetry. The examples in this paper are all from Python programs.

“…Single-valued neutrosophic sets were successfully applied to various decision making problems [3][4][5][6][7][8]. In addition, Zhang et al studied the neutrosophic logic algebras and discussed neutrosophic filters and neutrosophic triplet groups, which are the important foundation of the development of neutrosophic logic theory [9][10][11][12]. To facilitate research, "single-valued neutrosophic sets" are abbreviated as "neutrosophic sets" in this paper.…”

Section: Introductionmentioning

confidence: 99%

Neutrosophic Triangular Norms and Their Derived Residuated Lattices

2019

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Neutrosophic triangular norms (t-norms) and their residuated lattices are not only the main research object of neutrosophic set theory, but also the core content of neutrosophic logic. Neutrosophic implications are important operators of neutrosophic logic. Neutrosophic residual implications based on neutrosophic t-norms can be applied to the fields of neutrosophic inference and neutrosophic control. In this paper, neutrosophic t-norms, neutrosophic residual implications, and the residuated lattices derived from neutrosophic t-norms are investigated deeply. First of all, the lattice and its corresponding system are proved to be a complete lattice and a De Morgan algebra, respectively. Second, the notions of neutrosophic t-norms are introduced on the complete lattice discussed earlier. The basic concepts and typical examples of representable and non-representable neutrosophic t-norms are obtained. Naturally, De Morgan neutrosophic triples are defined for the duality of neutrosophic t-norms and neutrosophic t-conorms with respect to neutrosophic negators. Third, neutrosophic residual implications generated from neutrosophic t-norms and their basic properties are investigated. Furthermore, residual neutrosophic t-norms are proved to be infinitely ∨-distributive, and then some important properties possessed by neutrosophic residual implications are given. Finally, a method for producing neutrosophic t-norms from neutrosophic implications is presented, and the residuated lattices are constructed on the basis of neutrosophic t-norms and neutrosophic residual implications.Symmetry 2019, 11, 817 2 of 22 generalization of intuitionistic fuzzy sets, because their positive membership, neutral membership and negative membership are not independent completely. It is worth noting that picture fuzzy sets can be regarded as special neutrosophic sets [14,15], and also can be called standard neutrosophic sets [1,2,[16][17][18][19].Fuzzy logic plays a vital role in fuzzy set theory. T-norms, t-conorms, negators and implications are very important fuzzy logic operators. T-norms were originally defined by Menger [20], and then Schweizer and Sklar [21,22] redefined the t-norms which have been used to today. From the perspective of fuzzy logic, t-norms are the extension of intersection operation of fuzzy sets [23]. The algebraic properties of t-norms, for example, continuity, archimedean, strict, nilpotent and so on, are discussed in some papers [24][25][26][27][28]. Hu et al. studied t-norm extension operations [29]. Wang et al. discussed the lattice structure of algebra of fuzzy values [30]. The t-norms which satisfy the residual principle are an important class of t-norms, because they can produce fuzzy implications and constitute the residuated lattices [31][32][33][34]. Type-2 t-norms (t-conorms) and their residual operators on type-2 fuzzy sets were investigated by Li [25], which promote the development of fuzzy reference system. Intuitionistic fuzzy t-norms on intuitionistic fuzzy sets (L * -fuzzy sets) were proposed by Deschrij...