Hee Sik Kim

• Doctor of Philosophy
• Professor (Full) at Hanyang University

In this paper, we continue the investigation started in [1]. We obtain new results derived from novel concepts developed in analogy with others already established, e.g., the fact that leftoids (X, *) for φ are super-transitive if and only if φ(φ(x)) = φ(x) for all x ∈ X. In addition we apply fuzzy subsets in this context and we derive a number of...

By considering the notion of K-nonvanishing on a field K, we define polynomially defined d-ringoids, and obtain some conditions for it to be a polynomially defined strong d-ringoid.

In this paper, we discuss some iterations and apply it to $d/BCK$-algebras and semigroups. Given a $BCK$-algebra $(X,*, 0)$, we construct an edge $d$-algebra by using the notion of iterations.

In this paper, we introduce the notion of generalized anti-symmetry laws in groupoids, and we apply this concept to several algebraic structures. Moreover, we show that every Fibonacci sequence on ( C , *) is periodic.

To investigate the filter and deductive system of the Schaefer stroke Hilbert algebra using the Dokdo structure, the concept of Dokdo filter and Dokdo deductive system is defined, examples are given, and various properties are investigated. The Dokdo filter is formed by attaching appropriate conditions to the given Dokdo structure. The characteriza...

In this paper, for getting more results in groupoids, we consider a set and introduce the notion of a right (left) independent subset of a groupoid, and it is studied in detail. As a corollary of these properties, the following important result is proved: for any groupoid, there is a maximal right (left) independent subset. Moreover, the notion of...

In this paper, we introduce the notion of a disturbed Fibonacci sequence and obtain several properties related to Fibonacci sequences and apply the notion to Fibonacci rabbit farm to adjust the numbers of Fibonacci rabbits. AMS Subject Classification: Primary; 11B39, 39A05; secondary 39A22

Right feeble groups are defined as groupoids $(X,*)$ such that (i) $x, y\in X$ implies the existence of $a, b \in X$ such that $a*x = y$ and $b*y = x$. Furthermore, (ii) if $x, y, z \in X$ then there is an element $w\in X$ such that $x*(y*z) = w*z$. These groupoids have a "remnant" group structure, which includes many other groupoids. In this paper...

We introduce the notion of a left-twisted ring, and we construct a left-zero ring which is not a ring. We show that such a left-twisted ring does not have an identity. Also, we show that every non-zero element of the left-twisted ring is a pseudo unit of it.

The concept of BE-ringoid is introduced, and some properties of it are investigated. We discuss on a distributive BE-ringoid. Further, transitive BE-ringoid is defined and the relationship between distributive BE-ringoids and transitive BE-ringoids is considered, and we prove that every distributive BE-ringoid is a transitive BE-ringoid, but the co...

In this paper, we introduce the notion of a ringoid, and we obtain left distributive ringoids over a field which are not rings. We introduce several different types of ringoids, and also we discuss on (r, s)-ringoids. Moreover, we discuss geometric observations of the parallelism of vectors in several ringoids.

In this paper, we introduce the notions of a left and a right idenfunction in a groupoid by using suitable functions, and we apply this concept to several algebraic structures. Especially, we discuss its role in linear groupoids over a field. We show that, given an invertible function $ \varphi $, there exists a groupoid such that $ \varphi $ is a...

In this paper, we introduce the notion of a function kernel which was motivated from the kernel in group theory, and we apply this notion to several algebraic structures, e.g., groups, groupoids, BCK-algebras, semigroups, leftoids. Using the notions of left and right cosets in groupoids, we investigate some relations with function kernels. Moreover...

Given a fuzzy subgroup $\mu$ of a group $G$‎, ‎$x\rhd_uy$ if and only if $\mu(xy) < \mu(yx)$ defines a directed relation with an associated digraph $(G‎, ‎\rhd_u)$‎. ‎We consider $(\mu‎, ‎\nu)$-homomorphisms $\varphi‎: ‎(G‎, ‎\mu)\to (H‎, ‎\nu)$ where $\mu$ and $\nu$ are fuzzy subgroups of $G$ and $H$ respectively and the preservation of properties...

In this paper, we introduce and study the notion of a [Formula: see text]positive implicative[Formula: see text] left ray in groupoids, and we show that every normal subgroup of a group is a left ray of a group, and in every finite group, left rays are normal subgroups. Further, left absorptive subsets of groupoids are discussed and several example...

Background and purpose: Accurate radiologic prediction of cavernous sinus invasion by pituitary adenoma remains challenging. We aimed to assess whether 1-mm-slice-thickness MRI with deep learning-based reconstruction can better predict cavernous sinus invasion by pituitary adenoma preoperatively and to estimate the depth of invasion and degree of...

We extend the notions of right (left) independency and absorbent from groupoids to Bin(X) as a semigroup of all the groupoids on a set X and study and investigate many of their properties. We show that these new concepts are different by presenting several examples. In general, the concept of right (left) independence is a generalization and altern...

The concept of a neutrosophic set, which is a generalization of an intuitionistic fuzzy set and a para consistent set etc., was introduced by F. Smarandache. Since then, it has been studied in various applications. In considering a generalization of the neutrosophic set, Mohseni Takallo et al. used the interval valued fuzzy set as the indeterminate...

Background and purpose: The microenvironment of lymphomas is known to be highly variable and closely associated with treatment resistance and survival. We tried to develop a physiologic MR imaging-based spatial habitat analysis to identify regions associated with treatment resistance to facilitate the prediction of tumor response after initial che...

In this paper, we define the notions of intuitionistic fuzzy filters and intuitionistic fuzzy implicative (positive implicative, fantastic) filters on hoops. Then we show that all intuitionistic fuzzy filters make a bounded distributive lattice. Also, by using intuitionistic fuzzy filters we introduce a relation on hoops and show that it is a congr...

In this paper, we define a new algebraic structure known as a generalized pseudo BE-algebra which is a generalization of a pseudo BE-algebra. We construct some examples in order to show the existence of the generalized pseudo BE-algebra. Moreover, we characterize different classes of generalized pseudo BE-algebras by some results.

The concept of a commutative soju ideal in a BCK-algebra and a BCI-algebra is introduced, and their properties are investigated. The relationship between a soju ideal and a commutative soju ideal are discussed, and examples to show that any soju ideal may not be a commutative soju ideal are provided. Conditions for a soju ideal to be a commutative...

The BCK-neighborhood systems in d-algebras as measures of distance of these algebras from BCK-algebras is introduced. We consider examples of various cases and situations related to the general theory, as well as a compilcated analytical example of one of particular interest in the theory of pseudo-BCK-algebras. It appears also that a digraph theor...

Background and purpose: Differentiating glioblastoma from solitary brain metastasis preoperatively using conventional MR images is challenging. Deep learning models have shown promise in performing classification tasks. The diagnostic performance of a deep learning-based model in discriminating glioblastoma from solitary brain metastasis using pre...

Background and purpose: Currently available perfusion parameters are limited in differentiating early tumor progression and pseudoprogression with no insight about vessel size and type. We aimed to investigate differences in vessel size and type between early tumor progression and pseudoprogression in posttreatment glioblastoma and to demonstrate...

Background and purpose: Differences in molecular properties between one-molar and half-molar gadolinium-based contrast agents are thought to affect parameters obtained from dynamic contrast-enhanced imaging. The aim of our study was to investigate differences in dynamic contrast-enhanced parameters between one-molar nonionic gadobutrol and half-mo...

In this paper, we introduce the notion of a BV-algebra, and we show that a BV-algebra is logically equivalent to several algebras, i.e., BM-algebras, BT-algebras, BO-algebras and 0-commutative B-algebras. Moreover, we show that a BV-algebra with (F) is logically equivalent to several algebras, and we show some relationships between a BV-algebra wit...

In this paper, we introduce a version of the Moebius function and other special functions on a particular class of intervals for groupoids, and study them to obtain results analogous to those obtained in the usual lattice, combinatorics and number theory setting, but of course much more general due to the viewpoint taken in this paper.

In 2020, Kang et al. introduced the concept of a multipolar intuitionistic fuzzy set of finite degree, which is a generalization of a k-polar fuzzy set, and applied it to a BCK/BCI-algebra. The specific purpose of this study was to apply the concept of a multipolar intuitionistic fuzzy set of finite degree to a hyper BCK-algebra. The notions of the...

In this paper, we discuss the notion of an m-polar fuzzy (normal) subalgebra in B-algebras and its related properties. We consider characterizations of an m-polar fuzzy (normal) subalgebra. We define the concept of an m-polar intuitionistic fuzzy (normal) subalgebra in a B-algebra, and we characterize the m-polar intuitionistic fuzzy (normal) subal...

In this paper, as an extension of CI-algebras, we discuss the new notions of Neutro-CI-algebras and Anti-CI-algebras. First, some examples are given to show that these definitions are different. We prove that any proper CI-algebra is a Neutro-BE-algebra or Anti-BE-algebra. Also, we show that any NeutroSelf-distributive and AntiCommutative CI-algebr...

In this paper, as an extension of CI-algebras, we discuss the new notions of Neutro-CI-algebras and Anti-CI-algebras. First, some examples are given to show that these definitions are different. We prove that any proper CI-algebra is a Neutro-BE-algebra or Anti-BE-algebra. Also, we show that any NeutroSelf-distributive and AntiCommutative CI-algebr...

In this paper we define the notions of a norm and a prenorm on BL -algebras and we give some methods for constructing them. We introduce (pre)metrics on BL -algebras and find some connections between them and (pre)norms. Using prenorms, we find some conditions that a BL -algebra become a metric space. Finally, we show that if there exists a prenorm...

In this paper, we generalize the notion of an implicativity discussed in B C K -algebras, and apply it to some groupoids and B C K -algebras. We obtain some relations among those axioms in the theory of groupoids.

In this paper, we define the notion of a probability function on a poset which is similar to the probability function discussed on d-algebras, and obtain three probability functions on posets. Moreover, we define a probability realizer of a poset, and we provide some examples to describe its role for the standard probability function. We apply the...

A fuzzy set is an extension of an existing set using fuzzy logic. Soft set theory is a generalization of fuzzy set theory. Fuzzy and soft set theory are good mathematical tools for dealing with uncertainty in a parametric manner. The aim of this article is to introduce the concept of makgeolli structures using fuzzy and soft set theory and to apply...

In this paper, we define the concepts of ( ∈ , ∈ ) and ( ∈ , ∈ ∨ q ) -fuzzy filters of hoops, discuss some properties, and find some equivalent definitions of them. We define a congruence relation on hoops by an ( ∈ , ∈ ) -fuzzy filter and show that the quotient structure of this relation is a hoop.

In this paper, we introduce the notions of an almost φ -fuzzy subalgebra and an almost φ -fuzzy semi-ideal in groupoids by using the concept of a function. This concepts are generalizations of a fuzzy subalgebra and a fuzzy ideal in groupoids respectively. We discuss some relations between almost φ -fuzzy subalgebras and vanishing, level subsets. B...

In this paper, we investigate the graph structures on hoop algebras. First, by using the quasi-filters and r-prime (one-prime) filters, we construct an implicative graph and show that it is connected and under which conditions it is a star or tree. By using zero divisor elements, we construct a productive graph and prove that it is connected and bo...

In this paper, we introduce the notions of generalized commutative laws in algebras, and investigate their relations by using Smarandache disjointness. Moreover, we show that every pre-commutative B C K -algebra is bounded.

In this paper, we consider a theory of elements u of a groupoid ( X , * ) that are associated with certain functions u ^ : X → X , pseudo-inverse functions, which are generalizations of the inverses associated with units of groupoids with identity elements. If classifying the elements u as special of one of twelve types, then it is possible to do a...

If an algebra of type $A$ contains a subalgebra which is also an algebra of type $B$, then it is a Smarandache $B$-type $A$-algebra provided the subalgebra of type $B$ contains at least two elements. This generalizes the notion of Smarandache group, where the group of order $\geq 2$ is a subsemigroup of a semigroup. In this paper we investigate a n...

There are several equivalent axioms, which can be used to characterize the positive implicativity in B C K -algebras. In this paper, we investigate interrelationships among such axioms in a more general setting of groupoids, and several aspects regarding their differences in the theory of groupoids.

Atanassov's intuitionistic fuzzy sets extend the notion of fuzzy sets. In addition to Zadeh's membership function, a non-membership function is also considered. Intuitionistic fuzzy values play a crucial role in both theoretical and practical progress of intuitionistic fuzzy sets. This study introduces and explores various types of centroid transfo...

In this paper, we investigate the effects of certain variants of the commutative laws on properties of several families of algebras which are in general not commutative, such as groups, linear algebras, or actually quite anti-commutative, such as $BCK$-algebras and $d$-algebras among others. From results obtained it becomes clear that by considerin...

BACKGROUND AND PURPOSE: A small subset of primary central nervous system lymphomas exhibits high cerebral blood volume, which is indistinguishable from that in glioblastoma on dynamic susceptibility contrast MR imaging. Our study aimed to test whether estimates of combined perfusion and vascular permeability metrics derived from DSC-MR imaging can...

In this paper, we discuss some relations between the semigroup, Bin(X), of all groupoids (X,∗) and graphs. We discuss mimimum (mutual) covering sets in several groupoids and discuss distances of graphs with groupoids. Finally, we obtain some results on frame graphs with groupoids.

Background and purpose: A small subset of primary central nervous system lymphomas exhibits high cerebral blood volume, which is indistinguishable from that in glioblastoma on dynamic susceptibility contrast MR imaging. Our study aimed to test whether estimates of combined perfusion and vascular permeability metrics derived from DSC-MR imaging can...

In this paper, we observe that if X is a set and (Bin(X), □) is the semigroup of binary systems on X, then its center ZBin(X) consists of the locally-zero-semigroups and that these can be modeled as (simple) graphs and conversely. Using this device we show that we may obtain many results of interest concerning groupoids by reinterpreting graph theo...

In this paper, we de ne and study a function F : [0,∞) → R and extensions F: R→C, F : C → C which are continuous and such that if n ∈ Z, the set of all integers, then F(n)=Fn, the nth Fibonacci number based on F0= F1 = 1. If x is not an integer and x < 0, then F(x) may be a complex number, e.g., F(-1.5) = 1/2+i. If z = a + bi, then e F(z) = F(a) +...

In this paper analogs of Fibonacci sequences and Fibonacci numbers as well as Fibonacci functions (the case n = 2) for cases n = 3, 4. … are introduced. It is shown that these analogs are related to each other in a regular manner and that if (Formula presented) for a Fibonacci k-sequence, then (Formula presented) Many identities of types similar to...

In this paper, we introduce the notions of Fibonacci (co-)derivative of real-valued functions. We find general solutions of the equations Δ(f(x)) = g(x) and (Δ+ I)(f(x)) = g(x).

The notion of positive implicative superior ideals of BCK-algebras is introduced, and their properties are investigated. Relations between a superior ideal and a positive implicative superior ideal in BCK-algebras are studied, and conditions for a superior ideal to be a positive implicative superior ideal are provided. Characterizations of positive...

We discuss quotient structures of BCI-algebras using fuzzy ideals, i.e., if μ (resp. ν) is a fuzzy ideal in a BCI-algebra X (resp. Y), then X×Y μ×ν≅X/μ×Y/ν. Moreover, if J is an ideal of X such that J/μ is an ideal of X/μ, then X/μ J/μ≅X/J.

N this paper we define an algebra of type (2,2,0) associated with posets called a poring and we study several properties of porings and the linear algebras having such porings as their bases. In particular, we show that if (P; ∗, ·) is a standard poring then the distributive law (x · y) ∗ z = (x∗ z) · (y∗ z) holds iff the poset P (or the poset X) i...

In this paper we introduce a class of algebras whose bases over a field K are pogroupoids. We discuss several properties of these algebras as they relate to the structure of their associated pogroupoids and through these to the associated posets also. In particular the Jacobi form is 0 precisely when the pogroupoid is a semigroup, precisely when th...

In this paper, we solve the additive (α, β)-functional equation f(x) + f(y) + 2f(z) = αf(β(x + y + 2z)), (0.1) where α, β are fixed real or complex numbers with a ≠ 4 and αβ = 1. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the additive (α, β)-functional equation (0.1) in Banach spaces.

In this paper, the authors extend the theory of groupoids already developed for semigroups (Bin (X), □) in a growing number of research papers with X a set and Bin (X) the set of groupoids (binary systems) defined on X to the generalizations: fuzzy (sub)groupoids (largely known as used here) and hyperfuzzy (sub)groupoids (largely novel as used here...

In this paper, we study mappings f, g : X -> P, where P is a poset and X is a set, under the relation f parallel to g, of right parallelism, f (a) <= f (b) implies g (a) <= g(b), which is reflexive and transitive but not necessarily symmetric. We prove several results of the type: if f has property P and f parallel to g, then g has property P as we...

In this paper, we introduce a new algebra, called a BI-algebra, which is a generalization of a (dual) implication algebra and we discuss the basic properties of BI-algebras, and investigate ideals and congruence relations.

In this paper, we introduce and explore suggested notions of ‘above’, ‘below’ and ‘between’ in general groupoids, Bin(X), as well as in more detail in several well-known classes of groupoids, including groups, semigroups, selective groupoids (digraphs), d/BCK-algebras, linear groupoids over fields and special cases, in order to illustrate the usefu...

A characterization of int-soft ideal is considered, and an int-soft ideal generated by a soft set is discussed. A new int-soft ideal from old one is constructed.

In this paper, we introduce the concept of several types of groupoids related to semigroups, viz., twisted semigroups for which twisted versions of the associative law hold. Thus, if \((X,*)\) is a groupoid and if \(\varphi : X^2\rightarrow X^2\) is a function \(\varphi (a, b) = (u, v)\), then \((X,*)\) is a left-twisted semigroup with respect to \...

In this paper, we introduce fuzzy rank functions for groupoids, and we investigate their roles in the semigroup of binary systems by using the notions of right parallelisms and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgr...

The notions of hesitant fuzzy translations and hesitant fuzzy extensions of a hesitant fuzzy set on BCK/BCIalgebras are introduced, and related properties are investigated. We prove that every hesitant fuzzy translation of a hesitant fuzzy subalgebra (ideal) is a hesitant fuzzy subalgebra (ideal). Conditions for a hesitant fuzzy set to be a hesitan...

The notions of soft saturated values and soft dried values are introduced, and their applications in BCK/BCI-algebras are discussed. Using these notions, properties of energetic subsets are investigated. Using the concepts of intersectional (union) ideals, properties of right vanished (stable) subsets are explored.

In this paper we show how certain metabelian groups can be found within polynomial evaluation groupoids.

As a generalization of Q-algebra, the notion of pseudo Q-algebra is introduced, and some of their properties are investigated. The notions of pseudo subalgebra, pseudo ideal, and pseudo atom in a pseudo Q-algebra are introduced. Characterizations of their properties are provided.

We investigate the radical structure of a fuzzy polynomial ideal induced by a fuzzy ideal of a ring and study its properties. Given a fuzzy ideal β of R and a homomorphism f : R → R ′ , we show that if f x is the induced homomorphism of f , that is, f x ( ∑ i = 0 n a i x i ) = ∑ i = 0 n f ( a i ) x i , then f x...

In this paper, we generalize the left-zero semigroup by introducing. two different algebras, called a weak-zero groupoid and an (X, N)-zero groupoid, respectively and describe some properties related to Bin(X).1 Moreover, we fuzzify the notion of a weak -zero groupoid.

Molodtsov introduced the concept of soft set as a new mathematical tool for dealing with uncertainties. Recently, Cagman and Enginoglu [16] provided new definitions and operations on soft set theory. The paper applies new definitions and operations of soft sets to non-commutative residuated lattices. The notion of soft non-commutative residuated la...

By means of Dubois and Prade's idea, we apply rough fuzzy sets and fuzzy rough sets to algebraic structures. The concepts of rough fuzzy strong h-ideals (rough fuzzy prime ideals) and fuzzy rough strong h-ideals (fuzzy rough prime ideals) of hemirings are introduced, respectively. The relationships between them are investigated. Some characterizati...

In this paper, the concepts of M-fuzzy h-interior ideals and prime M-fuzzy h-ideals in M-Γ-hemirings are introduced. Some new properties of these kinds of M-fuzzy h-ideals are also given. Finally, some characterizations of the h-semisimple M-Γ-hemirings are investigated by these kinds of M-fuzzy h-ideals.

The concepts of (ε ε, qδ0)-fuzzy subalgebras and ε, V qδ0 -level sets are introduced, and related properties are investigated. Relations be- tween an (ε ε)-fuzzy subalgebra and an (ε ε V qδ0 )-fuzzy subalgebra are discussed, and characterizations of (ε ε V qδ0 )-fuzzy subalgebras are discussed. Homomorphic image and pre-image of an (ε ε V qδ0)-fuzz...

In this paper we discuss Fibonacci functions using the (ultimately) periodicity and we also discuss the exponential Fibonacci functions. Especially, given a non-negative real-valued function, we obtain several exponential Fibonacci functions. MSC: 11B39, 39A10.

Molodtsov’s soft set theory provides a general mathematical framework for dealing with uncertainty. The concepts of ( M , N ) - SI implicative (Boolean) filters of BL -algebras are introduced. Some good examples are explored. The relationships between ( M , N ) - SI filters and ( M , N ) - SI implicative filters are discussed. Some properties...

We define a ranked trigroupoid as a natural followup on the idea of a ranked bigroupoid. We consider the idea of a derivation on such a trigroupoid as representing a two-step process on a pair of ranked bigroupoids where the mapping d is a self-derivation at each step. Following up on this idea we obtain several results and conclusions of interest....

The notions of falling fuzzy implicative, positive implicative and fantastic filters of a BL-algebra are introduced based on the theory of falling shadows and fuzzy sets. The relations between fuzzy implicative, positive implicative and fantastic filters and falling fuzzy implicative, positive implicative and fantastic filters are provided. Finally...

The notions of int-soft semigroups and int-soft left (resp., right) ideals are introduced, and several properties are investigated. Using these notions and the notion of inclusive set, characterizations of subsemigroups and left (resp., right) ideals are considered. Using the notion of int-soft products, characterizations of int-soft semigroups and...

We discuss properties of a class of real-valued functions on a set X (2) constructed as finite (real) linear combinations of functions denoted as [(X, ∗); μ ], where (X, ∗) is a groupoid (binary system) and μ is a fuzzy subset of X and where [(X, ∗); μ ](x, y): = μ (x∗y) - min⁡{ μ (x), μ (y)}. Many properties, for example, μ being a fuzzy subgroupo...

The purpose of this paper is to give a foundation for providing a new soft algebraic tool in considering many problems containing uncertainties. In order to provide these new soft algebraic structures, we discuss a new soft set-(M, N)-soft intersection set, which is a generalization of soft intersection sets. We introduce the concepts of (M, N)-SI...

In this paper we introduce the notion of a fuzzy polynomial ideal alpha(x) of a polynomial ring R[x] induced by a fuzzy ideal alpha of a ring R, and obtain an isomorphism theorem of a ring of fuzzy cosets of alpha(x). It is shown that a fuzzy ideal alpha of a ring is fuzzy prime if and only if alpha(x) is a fuzzy prime ideal of R[x]. Moreover, we s...

In this paper, we introduce the notion of generalized Fibonacci sequences over a groupoid and discuss it in particular for the case where the groupoid contains idempotents and pre-idempotents. Using the notion of Smarandache-type P-algebra, we obtain several relations on groupoids which are derived from generalized Fibonacci sequences. MSC: 11B39...

We introduce the notion of abelian fuzzy subsets on a groupoid, and we observe a variety of consequences which follow. New notions include, among others, diagonal symmetric relations, several types of quasi orders, convex sets, and fuzzy centers, some of whose properties are also investigated.

In this paper we introduce the notion of TN-groupoids in an arbitrary groupoid, and discuss some properties in Bin(X), and obtain several results related with non-negativity of norms.

In this paper, we introduce a BN-algebra, and we prove that a BN-algebra is 0-commutative, and an algebra X is a BN-algebra if and only if it is a 0-commutative BF-algebra. And we introduce a quotient BN-algebra, and we investigate some relations between BN-algebras and several algebras.

In this paper we introduce for an arbitrary algebra (groupoid, binary system) (X; *) a sequence of algebras (X; *)n = (X; ∘), where x ∘ y = [x * y]n = x * [x * y]n−1, [x * y]0 = y. For several classes of examples we study the cycloidal index (m, n) of (X; *), where (X; *)m = (X; *)n for m > n and m is minimal with this property. We show that (X; *)...

In this paper, for a given groupoid X, * and a relation R on X we consider a related structure LFRX = {< x, y, z > ∈ X3 | xRy, yRz}, and we discuss three different types on LFRX, and we construct a linear fuzzy real BCK-algebra LF≤X, $\ominus$, ε0 and a new BCK-algebra LF≤X, $\oast$, ε0 from a BCK-algebra X, *, 0.

In this paper, we introduce the notion of ranked bigroupoids and we define as well as discuss (X, *, ω)-self-(co)derivations. In addition we define rankomorphisms and (X, *, ω)-scalars for ranked bigroupoids, and we consider some properties of these as well.

An A-semiring has commutative multiplication and the property that every proper ideal B is contained in a prime ideal P, with √B, the intersection of all such prime ideals. In this paper, we define homogeneous ideals and their radicals in a graded semiring R. When B is a proper homogeneous ideal in an A-semiring R, we show that √B is homogeneous wh...

We explore properties of the set of d-units of a d-algebra. A property of interest in the study of d-units in d-algebras is the weak associative property. It is noted that many other d-algebras, especially BCK-algebras, are in fact weakly associative. The existence of d/BCK-algebras which are not weakly associative is demonstrated. Moreover, the no...

In this paper we consider Fibonacci functions on the real numbers R , i.e. , functions f : R → R such that for all x ∈ R , f ( x + 2 ) = f ( x + 1 ) + f ( x ) . We develop the notion of Fibonacci functions using the concept of f -even and f -odd functions. Moreover, we show that if f is a Fibonacci function then lim x → ∞ f ( x + 1 ) f ( x ) = 1 +...