BCC-algebras and residuated partially-ordered groupoid

The concept of basic implication algebra (BI-algebra) has been proposed to describe general non-classical implicative logics (such as associative or non-associative fuzzy logic, commutative or non-commutative fuzzy logic, quantum logic). However, this algebra structure does not have enough characteristics to describe residual implications in-depth, so we propose a new concept of strong BI-algebra, which is exactly the algebraic abstraction of fuzzy implication with pseudo-exchange principle (PEP) property. Furthermore, in order to describe the characteristics of the algebraic structure corresponding to the non-commutative fuzzy logics, we extend strong BI-algebras to the non-commutative case, and propose the concept of pseudo-strong BI (SBI)-algebra, which is the common extension of quantum B-algebras, pseudo-BCK/BCI-algebras and other algebraic structures. We establish the filter theory and quotient structure of pseudo-SBI- algebras. Moreover, based on prequantales, semi-uninorms, t-norms and their residual implications, we introduce the concept of residual pseudo-SBI-algebra, which is a common extension of (non-commutative) residual lattices, non-associative residual lattices, and also a special kind of residual partially-ordered groupoids. Finally, we investigate the filters and quotient algebraic structures of residual pseudo-SBI-algebras, and obtain a unity frame of filter theory for various algebraic systems.

show abstract

“…Using the property of residuated lattice-ordered groupoid (see [20,30]), y → (z ⊗ y) ≤ x → (z ⊗ y). From (2), there exists z ≤ y → (z ⊗ y).…”

Section: Definition 20mentioning

confidence: 99%

“…(4) ∀x, y ∈ X. By Proposition 22(2), y ≤ x (x ⊗ y), and because of Proposition 22(3), there exists The literature [30] has listed many properties of residuated partially-ordered groupoid. Using Proposition 21, residuated pseudo-SBI-algebra is a special residuated partially-ordered groupoid.…”

Section: Definition 20mentioning

confidence: 99%

Filters in Strong BI-Algebras and Residuated Pseudo-SBI-Algebras

Zhang

Wang

2020

Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…As a generalization of BCI-algebra, Dudek and Jun [16] introduced the notion of pseudo-BCI algebras. Moreover, pseudo-BCI algebra is also as a generalization of pseudo-BCK algebra (which has a close connection with various noncommutative fuzzy logic formal systems; see [17][18][19][20][21][22][23][24]). For nonclassical logic algebra systems, the theory of filters (ideals) plays an important role (see [25][26][27][28][29][30][31][32][33]).…”

Section: Introductionmentioning

confidence: 99%

Neutrosophic Filters in Pseudo-Bci Algebras

Zhang

Mao

et al. 2018

Int. J. UncertaintyQuantification

Self Cite

View full text Add to dashboard Cite

The concept of the neutrosophic set was introduced by Smarandache; it is a mathematical tool for handling problems involving imprecise, indeterminacy and inconsistent data. The notion of pseudo-BCI algebra was introduced by Dudek and Jun; it is a kind of nonclassical logic algebra and has a close connection with various noncommutative fuzzy logics. In this paper, neutrosophic set theory is applied to pseudo-BCI algebras. The new concepts of neutrosophic filter, neutrosophic normal filter, antigrouped neutrosophic filter, and neutrosophic p-filter in pseudo-BCI algebras are proposed, and their basic properties are presented. Moreover, by using the concept of (alpha, beta, gamma)-level set in neutrosophic sets, the relationships between fuzzy filters and neutrosophic filters are discussed.

show abstract

“…Jun introduced the concept of pseudo-BCI algebra in [12]. In fact, there are many other non-classical logic algebraic systems related to BCK-and BCI-algebras, such as BCC-algebra, BZ-algebra and so forth, some monographs and papers on these topics can be found in [7][8][9][13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

The Structure Theorems of Pseudo-BCI Algebras in Which Every Element is Quasi-Maximal

Zhang

2018

Symmetry

Self Cite

View full text Add to dashboard Cite

For mathematical fuzzy logic systems, the study of corresponding algebraic structures plays an important role. Pseudo-BCI algebra is a class of non-classical logic algebras, which is closely related to various non-commutative fuzzy logic systems. The aim of this paper is focus on the structure of a special class of pseudo-BCI algebras in which every element is quasi-maximal (call it QM-pseudo-BCI algebras in this paper). First, the new notions of quasi-maximal element and quasi-left unit element in pseudo-BCK algebras and pseudo-BCI algebras are proposed and some properties are discussed. Second, the following structure theorem of QM-pseudo-BCI algebra is proved: every QM-pseudo-BCI algebra is a KG-union of a quasi-alternating BCK-algebra and an anti-group pseudo-BCI algebra. Third, the new notion of weak associative pseudo-BCI algebra (WA-pseudo-BCI algebra) is introduced and the following result is proved: every WA-pseudo-BCI algebra is a KG-union of a quasi-alternating BCK-algebra and an Abel group.

show abstract

BCC-algebras and residuated partially-ordered groupoid

Cited by 14 publications

References 25 publications

Filters in Strong BI-Algebras and Residuated Pseudo-SBI-Algebras

Filters in Strong BI-Algebras and Residuated Pseudo-SBI-Algebras

Neutrosophic Filters in Pseudo-Bci Algebras

The Structure Theorems of Pseudo-BCI Algebras in Which Every Element is Quasi-Maximal

Contact Info

Product

Resources

About