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Fuzzy BCK-Algebras

Mahasin. A. Ahmed, Esmat A. Amhed
2020 Journal of Applied Mathematics and Physics  
The purpose of this paper is to give the fuzzification of BCK-algebras and we study relationships between BCK-filters and lattice filters in BCK-algebras.  ...  For the general development of BCK-algebras, the filter theory plays an important role as well as ideal theory.  ...  Conflicts of Interest The authors declare no conflicts of interest regarding the publication of this paper.  ... 
doi:10.4236/jamp.2020.85071 fatcat:i26j4bztd5an7p4cuts5gtoqm4

Extensions of BCK-algebras

A. Radfar, A. Rezaei, A. Borumand Saeid, Lishan Liu
2016 Cogent Mathematics  
In this paper, we introduce the notion of sBCI/sBCK/eBCI/eBCK-algebras as a generalization of the notion of BCI/BCK-algebras. This structure is studied in detail.  ...  , weak-R0 algebras, etc.) can be expressed as particular cases of BCK-algebras and the BCK-algebras are particular cases of BCI-algebras.  ...  Funding The authors received no direct funding for this research. Author details A. Radfar 1  ... 
doi:10.1080/23311835.2016.1265297 fatcat:3adnlkrsczczddera3llbzxeuu

State BCK-algebras and State-Morphism BCK-algebras [article]

R. A. Borzooei, A. Dvurečenskij, O. Zahiri
2013 arXiv   pre-print
In the paper, we define the notion of a state BCK-algebra and a state-morphism BCK-algebra extending the language of BCK-algebras by adding a unary operator which models probabilistic reasoning.  ...  In addition, we show the generators of quasivarieties of state-morphism BCK-algebras.  ...  Note that if (X, * , 0) is a BCK-algebra such that (X, ≤) is a lattice, it is called a BCK-lattice. Then by [29, Thm 2.2.6], X satisfies the identity (x * y) ∧ (y * x) = 0.  ... 
arXiv:1304.6963v1 fatcat:rdvu2lqoc5gahip7x46bi7tane

Trees as commutative BCK-Algebras

William H. Cornish
1981 Bulletin of the Australian Mathematical Society  
A new method of constructing commutative BCK-algebras is given. It depends upon the notion of a valuation of a lower semilattice in a given commutative BCK-algebra.  ...  Thus, any tree of finite height possesses a uniquely determined commutative BCK-structure.  ...  Corn i sh completely describe the lattice of subvarieties of the variety of commutative BCK-algebras satisfying the identity xy = xy Valuations Because of Yutani [75] , a commutative BCK-algebra can  ... 
doi:10.1017/s0004972700007024 fatcat:qxmcmdsbn5bv7amszikt5xkl2m

Quotient hyper pseudo BCK-algebras

Rajab Ali Boorzoei, Habib Harizavi, Tayebeh Koochakpoor
2013 Discussiones Mathematicae - General Algebra and Applications  
In this paper, we first investigate some properties of the hyper pseudo BCK-algebras.  ...  Then we define the concepts of strong and reflexive hyper pseudo BCK-ideals and establish some relationships among them and the other types of hyper pseudo BCK-ideals.  ...  Acknowledgements The authors are deeply grateful to the referees for the valuable suggestions and comments.  ... 
doi:10.7151/dmgaa.1205 fatcat:akilcrv66zgrxinidgo5livsc4

Fuzzy ideals of pseudo-BCK Algebras

Grzegorz Dymek, Andrzej Walendziak
2012 Demonstratio Mathematica  
AbstractCharacterizations of fuzzy ideals of a pseudo-BCK algebra are established. Conditions for a fuzzy set to be a fuzzy ideal are given. Given a fuzzy set  ...  The authors thank the referee for his/her remarks which were incorporated into this revised version.  ...  Given a fuzzy set µ, we make the least fuzzy ideal containing µ. This leads us to show that the set of fuzzy ideals of a pseudo-BCK algebra is a complete lattice.  ... 
doi:10.1515/dema-2013-0346 fatcat:cep2rrpb6jeephvl2dpty7gt3a


Anatolij Dvurečenskij
2000 Demonstratio Mathematica  
We show that the category of product BCK-algebras is categorically equivalent to the category of ¿-rings with special properties.  ...  We introduce a product on commutative BCK-algebras with the relative cancellation property, i.e., commutative BCK-algebras (X; *, 0) satisfying the condition for x,y,a G X, with a < x, a < y and x * a  ...  It follows ideas developed in above and in [BDG] , and from the observation that an element u of X is a quasi strong unit iff the ideal of X generated by u is equal to X. u Ideals in product BCK-algebras  ... 
doi:10.1515/dema-2000-0102 fatcat:kcw2jecf65f5tdr4rorn6n7t5e

On S-topological Bck-algebra

Alias Khalaf
2020 The Journal of The University of Duhok  
The aim of this paper is to introduce the concept of s- topological BCK-algebras which is a BCK-algebra equipped with a spe- cial type of topology that makes the operation defined on it s-topologically  ...  Deeba studied the topological aspects of the BCK-structure in a mode comparable to the study of topological groups, rings and lattices.  ...  Lee and Rio in [3] studied the topological structures making the star operation of BCK-algebra continuous and they found nice properties of such BCK-algebras.  ... 
doi:10.26682/csjuod.2020.23.1.21 fatcat:5fovajmpgzhfhi77bnsi4vz2wa


J. G. Raftery, T. Sturm
1996 Demonstratio Mathematica  
Recall that every 5CA'-algebra satisfies the identities For an element a of our BCK-algebra A, we denote by (a] the principal order ideal of the ordered set (v4;<) generated by a, i.e., (a] = {x 6 A :  ...  ) of BCK-algebras.  ... 
doi:10.1515/dema-1996-0113 fatcat:exv57c3fevhvrlqd2hq2acavba


Shunsuke Funakoshi
2013 Scientiae mathematicae Japonicae  
The following sections describe "BCK-Algebra" and "Non-Greek Mathematics," which are thought of as expressing K. Iséki's "mathematical ideas."  ...  The accomplishments of K. Iséki are spread over all fields of mathematics,and he has authored a large number of excellent and original research papers and books.  ...  (2) Research on theories such as lattice theory, which originated as a result of Dedekind's attempted algebraic (abstract) formulation of unions and intersections in the set theory during his research  ... 
doi:10.32219/isms.76.2_235 fatcat:lffdmexbdncuzbryr353hgb344

Subdirect products of totally ordered BCK-algebras

Isidore Fleischer
1987 Journal of Algebra  
the stimulus of their interest.  ...  ACKNOWLEDGMENTS I thank Teo Sturm for letting me have a preprint of [CST] as well as a number of other preprints and reprints not otherwise accessible to me, Denis A.  ...  "BCK-algebras" are structures axiomatized with a binary composition intended to mimic residuation in partially ordered monoids and, more particularly, set subtraction.  ... 
doi:10.1016/0021-8693(87)90224-9 fatcat:um5qdhdrtfgapo33dvwire2bua

Neutrosophic Positive Implicative N -Ideals in BCK-Algebras

Young Jun, Florentin Smarandache, Seok-Zun Song, Madad Khan
2018 Axioms  
The notion of a neutrosophic positive implicative N -ideal in BCK-algebras is introduced, and several properties are investigated.  ...  An extension property of a neutrosophic positive implicative N -ideal based on the negative indeterminacy membership function is discussed.  ...  Author Contributions: This paper is a result of common work of the authors in all aspects. Conflicts of Interest: The authors declare no conflict of interest  ... 
doi:10.3390/axioms7010003 fatcat:oelmcddynfas5mzbmo2ehbmwgi

Fuzzy hyper p-ideals of hyper BCK-algebras

Aslam Malik, Muhammad Touqeer
2015 Filomat  
The connections between "fuzzy (weak, strong) hyper p-ideals" are discussed and "the strongest fuzzy relation" on a "hyper BCK-algebra" is conferred.  ...  The paper is a reflection of "fuzzy sets" applied to "hyper p-ideals" and their comparison with simple "fuzzy hyper BCK-ideals".  ...  This paper confers, "the concept of fuzzification of (weak, strong) hyper p-ideals in hyper BCK-algebras" and associated properties.  ... 
doi:10.2298/fil1508769m fatcat:zjbi43apxrgnjahdsd572tn27a


Helena Bednarczyk
1984 Demonstratio Mathematica  
Then the knowledge of subdirectly irreducible positive implicative BCK-algebras gives us little information on the structure of these algebras.  ...  A non-empty subset A of a BCK-algebra X is said to be an ideal if and only if (iv) x * y 0 if x*y, x for the other cases 0 if xsy, x for the other cases (1) (2) x,y * x e A imply y e A It follows from  ... 
doi:10.1515/dema-1984-0408 fatcat:n2rqjgqngravraam7jf5a2j3re

Fuzzy Filter Spectrum of a BCK Algebra

Xiao Long Xin, Wei Ji, Xiu Juan Hua
2011 International Journal of Mathematics and Mathematical Sciences  
We prove that the set of all fuzzy s-prime filters of a bounded commutative BCK-algebra forms a topological space.  ...  Moreover, we show that the set of all fuzzy s-prime filters of a bounded implicative BCK-algebra is a Hausdorff space.  ...  Acknowledgment The paper is supported by National Nature Science Foundation of Shaanxi Province 2007A19 . International Journal of Mathematics and Mathematical Sciences  ... 
doi:10.1155/2011/795934 fatcat:4q7vv3p34jeubngogmaq3n65qm
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