Search

We generalize to d 0 -algebras a result of Riečanová about the decomposition of a D-lattice by means of a family of central elements. [ABSTRACT FROM AUTHOR]

The notion of strong fuzzy generalized exchange (GE)-filters and fuzzy GE-ideals of bordered GE-algebras is studied, and their related properties are investigated. We proved that the intersection of strong fuzzy GE-filters (fuzzy GE-ideals) is a strong fuzzy GE-filter (fuzzy GE-ideal). We studied the conditions for a fuzzy subset of a bordered GE-algebra to be a strong fuzzy GE-filter and the characterization of a strong fuzzy GE-filter. We studied that under some conditions, any fuzzy subset of a bordered GE-algebra can be a fuzzy GE-ideal. A given fuzzy GE-ideal and an element in a bordered fuzzy GE-algebra is established.

The (fuzzy) filter of the Sheffer stroke Hilbert algebra was addressed by Oner, Katican and Borumand Saeid. The weak version of the filter was discussed by Jun and Oner. In this manuscipt, with the fuzzy version of the weak filter in mind, the notions of fuzzy weak filters and (∈, ∈ ∨q)-fuzzy weak filters are introduced, and their properties are explored. Conditions under which t-level set, Qt-set and t-∈ ∨q-set become weak filters are explored in relation to fuzzy weak filters and (∈, ∈ ∨q)-fuzzy weak filters. The relationship and characterization of the fuzzy weak filter and the (∈, ∈ ∨q)-fuzzy weak filter are investigated.

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. [ABSTRACT FROM AUTHOR]

A new structure, so called soju structure, is introduced by combining intuitionistic fuzzy set and soft set, which is applied to BCK/BCI-algebras. The notions of soju subalgebra and soju ideal in BCK/BCI-algebras are introduced, and related properties are investigated. A strong soju subalgebra in BCI-algebras is introduced, and a condition for a soju subalgebra to be strong is provided. Relations between soju subalgebra and soju ideal are discussed, and conditions for a soju structure to be a soju ideal in a BCK-algebra are considered. Characterizations of soju subalgebra and soju ideal are considered, and the homomorphic image and preimage of soju subalgebra is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce commutative, medial, normal, irreducible and prime filters in eBE-algebras, and investigate some of them properties. We define and study the concept of commutative eBE-algebra. In particular, we show that every commutative eBE-algebra is a commutative BE-algebra and prove that each filter of such algebra is commutative. Moreover, we give a construction of quotient eBE-algebras via normal filters. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to characterize commutativity, B lo -homomorphism, and B lo -ideal of B lo -algebras. [ABSTRACT FROM AUTHOR]

In this article, we further study the filter theory of semihoops. Moreover, we use the prime (maximal) filters to construct the prime (maximal) spectrum on semihoops, and prove that the prime spectrum is a compact T0{T}_{0} topological space and that the maximal spectrum is a compact T2{T}_{2} topological space. As an application, in order to study the relationship between the spectrum of semihoops and the spectrum of lattices, we introduce the reticulations of semihoops and obtain some related results.

Using the results about maximal B p -subalgebras, we show that there are no noncommutative B-simple B-algebras of the following orders: (i) between 1 and 24, (ii) between 24 and 36, and (iii) between 36 and 60. [ABSTRACT FROM AUTHOR]

In this paper, we study projective algebras in varieties of (bounded) commutative integral residuated lattices. We make use of a well-established construction in residuated lattices, the ordinal sum, and the order property of divisibility. Via the connection between projective and splitting algebras, we show that the only finite projective algebra in FL ew is the two-element Boolean algebra. Moreover, we show that several interesting varieties have the property that every finitely presented algebra is projective, such as locally finite varieties of hoops. Furthermore, we show characterization results for finite projective Heyting algebras, and finitely generated projective algebras in locally finite varieties of bounded hoops and BL-algebras. Finally, we connect our results with the algebraic theory of unification. [ABSTRACT FROM AUTHOR]

In this paper, as a further generalization of m-polar fuzziness subalgebras, we present the conception of an m-polar [Formula: see text]-fuzzy subalgebra in BCK/BCI-algebras and discuss some interesting properties of it. We define an m-polar [Formula: see text]-fuzzy subalgebras and explore some of its significant attributes. Certain features of m-polar [Formula: see text]-fuzzy subalgebras are established.

The notions of an interior GE-filter, a weak interior GE-filter and a belligerent interior GE-filter are introduced, and their relations and properties are investigated. Example of a GE-filter which is neither an interior GE-filter nor a weak interior GE-filter is provided. Relations between a weak interior GE-filter and an interior GE-filter are discussed, and conditions under which every weak interior GE-filter is an interior GE-filter are investigated. Relations between a belligerent interior GE-filter and an interior GE-filter are displayed, and conditions for an interior GE-filter to be a belligerent interior GE-filter are considered. Given a subset and an element, an interior GE-filter is established, and conditions for a subset to be a belligerent interior GE-filter are discussed. The extensibility of the beligerant interior GE-filter is debated. Relationships between weak interior GE-filter and belligerent interior GE-filter of type 1, type 2 and type 3 are founded.

In this paper, the notions of n-fold positive implicative prefilter and n-fold implicative prefilter in EQ-algebras are introduced and several properties, characterizations and equivalent conditions are provided. It is proved that the quotient EQ-algebra induced by an n-fold positive implicative prefilter is n-idempotent. Also, it is proved that in an n-idempotent EQ-algebra, any filter is an n-fold positive implicative filter. In the sequel, we investigate the relationships between these two types of prefilters. Finally, some characterizations of n-fold implicative prefilters in bounded EQ-algebras are given.

We introduce the notion of ∧-projection in order to extend to d0-algebras the concept of Kalmbach measurable elements with respect to an outer measure μ. We prove, in case μ is faithful, that Kalmbach measurable ∧-projections are quasi-central, thus generalizing a result known for orthomodular lattices, and recently extended to D-lattices. [ABSTRACT FROM AUTHOR]

The concept of a (strong) set-valued BCK/BCI-morphism in BCK/BCI-algebras is considered, and several properties are investigated. Conditions for a set-valued mapping to be a set-valued BCK/BCI-morphism are given. Using the concept of generalized approximation space, generalized rough subalgebra (ideal) in BCK/BCI-algebras are introduced, and investigate their properties. Using the concept of generalized approximation space and ideal of BCK/bCI-algebra, another type of generalized lower and upper approximations based on the ideal is considered, and then several properties are investigated.

A new algebraic structure was introduced, called an eGE-algebra, which is a generalisation of a GE-algebra and investigated its properties. We explore the definition of filters and the quotient algebra associated with such filters.

In this paper, as a further generalization of m-polar fuzziness subalgebras, we present the conception of an m-polar (ω , θ) -fuzzy subalgebra in BCK/BCI-algebras and discuss some interesting properties of it. We define an m-polar (∈ γ ˆ , ∈ γ ˆ ∨ q δ ˆ ) -fuzzy subalgebras and explore some of its significant attributes. Certain features of m-polar (∈ γ ˆ , ∈ γ ˆ ∨ q δ ˆ ) -fuzzy subalgebras are established. [ABSTRACT FROM AUTHOR]

In this paper we study prime spectra of commutative BCK-algebras. We give a new construction for commutative BCK-algebras using rooted trees, and determine both the ideal lattice and prime ideal lattice of such algebras. We prove that the spectrum of any commutative BCK-algebra is a locally compact generalized spectral space which is compact if and only if the algebra is finitely generated as an ideal. Further, we show that if a commutative BCK-algebra is involutory, then its spectrum is a Priestley space. Finally, we consider the functorial properties of the spectrum and define a functor from the category of commutative BCK-algebras to the category of distributive lattices with zero. We give a partial answer to the question: what distributive lattices lie in the image of this functor? [ABSTRACT FROM AUTHOR]

The concept of forceable weak eGE-algebras is introduced, and its associated properties are investigated. The structure of GE-filters in the GE-algebras derived from weak eGE-algebras is discussed. The relationship between an eGE-algebra and a forceable weak eGE-algebra is considered, and the characterization of an eGE-filter in weak eGE-algebras is established. Using eGE-filter in forceable weak eGE-algebras, an equivalence relation is induced, and then the quotient set is constructed. A binary operation is given on the quotient set to create a quotient GE-algebra, and an example is given to illustrate it. [ABSTRACT FROM AUTHOR]

We provide some properties of maximal Bp-subalgebras of B-algebras. In particular, we show that for each prime p, a finite B-algebra has a maximal Bp-subalgebra. We also show that for a finite B-algebra of order prm, where (p, m) = 1, any two maximal Bp-subalgebras are conjugate and the number of maximal Bp-subalgebras is kp + 1 for some k ∈ 𝕑+.

The notion of a RM algebra, introduced recently, is a generalization of many other algebras of logic. The class of RM algebras contains (weak-)BCC algebras, BCH algebras, BCI algebras, BCK algebras and many others. A RM algebra is an algebra A = (A; →, 1) of type (2, 0) satisfying the identities: x → x = 1 and 1 → x = x. In this paper we study the set of maximal elements of a RM algebra, branches of a RM algebra and moreover translation deductive systems of a RM algebra giving so called the Representation Theorem for RM algebras.

In this paper, we introduced the concept of a soft hoop and we investigated some of their properties. Then, we established different types of intersections and unions of the family of soft hoops. We defined two operations ⊙ and → on the set of all soft hoops and we proved that with these operations, it is a hoop and also is a Heyting algebra. Finally we introduced a congruence relation on the set of all soft hoops and we investigated the quotient of it.

In this paper, we first introduce a generalized K-algebra (briefly, gK-algebra) (P , · , ⊙ , e , - 1) which is constructed on a non-symmetric polygroup (P , · , e , - 1) , by adjoining binary hyperoperation ⊙ defined by x ⊙ y = x · y - 1 , for all x , y ∈ P . Then, we show that every gK-algebra associated to a canonical hypergroup is a hyper B-algebra with constant e. Finally, we define generalized gK-subalgebras, (prime) generalized gK-ideals, homomorphisms of (P , · , ⊙ , e , - 1) and quotient generalized gK-algebras. [ABSTRACT FROM AUTHOR]

The new definitions of intuitionistic from magnified translation fuzzy in AT-subalgebra and AT-ideals on AT-algebras study the relationships between them. Also, introduce the homomorphism from fuzzy AT-subalgebras and fuzzy AT-ideals are investigated. [ABSTRACT FROM AUTHOR]

The relationship between a transitive GE-algebra and a belligerent GE-algebra (also, between an antisymmetric GE-algebra and a left exchangeable GE-algebra) is displayed. A condition for the trivial GE-filter to be a belligerent GE-filter is provided. The least GE-filter containing a given GE-filter and one element is formed. Conditions under which any set can be turned into a GE-filter are described. The notions of ⊙ -GE-algebras and prominent GE-filters are introduced, and their properties are investigated. The relationship between a prominent GE-filter and a GE-filter are considered, and conditions for a GE-filter and the trivial GE-filter to be a prominent GE-filter are given. The conditions under which the upset of an element will be a prominent GE-filter are examined, and the extension property for the prominent GE-filter is established. The notion of GE-morphism is introduced, and the GE-morphism theorem is considered. Conditions for the kernel of a GE-morphism to be a belligerent GE-filter are provided, and the condition that if the kernel of a GE-morphism is a belligerent GE-filter, then the trivial GE-filter becomes a belligerent GE-filter is given. A condition for the kernel of a GE-morphism to be a belligerent GE-filter is discussed, and the vice versa is also considered. [ABSTRACT FROM AUTHOR]

The notion of an m-polar (∈ , ∈) -fuzzy a-ideal is introduced, and its properties are investigated. The relationship between m-polar fuzzy subalgebra, m-polar fuzzy ideal, and m-polar (∈ , ∈) -fuzzy a-ideal is examined. Conditions for an m-polar fuzzy ideal to be an m-polar (∈ , ∈) -fuzzy a-ideal are provided. The relationship between m-polar (∈ , ∈) -fuzzy p-ideal, m-polar (∈ , ∈) -fuzzy q-ideal, and m-polar (∈ , ∈) -fuzzy a-ideal is shown. The normal m-polar (∈ , ∈) -fuzzy a-ideal is introduced, and its characterizations are considered. Characterizations and extension property of an m-polar (∈ , ∈) -fuzzy a-ideal are discussed. [ABSTRACT FROM AUTHOR]

Injective pseudo-BCI algebras are studied. There is shown that the only injective pseudo-BCI algebra is the trivial one.

The notion of MBJ-neutrosophic ideal is introduced, and its properties are investigated. Conditions for an MBJ-neutrosophic set to be an MBJ-neutrosophic ideal are provided. In a BCK/BCI-algebra, a condition for an MBJ-neutrosophic set to be an MBJ-neutrosophic ideal is given. In a BCK-algebra, a condition for an MBJ-neutrosophic subalgebra to be an MBJ-neutrosophic ideal is given. In a BCI-algebra, conditions for an MBJ-neutrosophic ideal to be an MBJ-neutrosophic subalgebra are considered. In an (S)-BCK-algebra, we show that every MBJ-neutrosophic ideal is an MBJ-neutrosophic ∘-subalgebra, and a characterization of an MBJ-neutrosophic ideal is established.

In this paper, the notion of closed cubic intuitionistic ideals, cubic intuitionistic p-ideals and cubic intuitionistic a-ideals in BCI-algebras are introduced, and several related properties are investigated. Relations between cubic intuitionistic subalgebras, closed cubic intuitionistic ideals, cubic intuitionistic q-ideals, cubic intuitionistic p-ideals and cubic intuitionistic a-ideals are discussed. Conditions for a cubic intuitionistic ideal to be a cubic intuitionistic p-ideal are provided. Characterizations of a cubic intuitionistic a-ideal are considered. The cubic intuitionistic extension property for a cubic intuitionistic a-ideal is established.

In this paper, we introduce the notion of state maps from a semihoop H1 to another semihoop H2, which is a generalization of internal states (or state operators) on a semihoop H. Also we give a type of special state maps from a semihoop H1 to H1, which is called internal state maps (or IS-maps). Then we give some examples and basic properties of (internal) state maps on semihoops. Moreover, we discuss the relations between state maps and internal states on other algebras. Then we introduce several kinds of filters by state maps on semihoops, called SM-filters, state filters and dual state filters, respectively, and discuss the relations among them. Furthermore we introduce and study the notion of prime SM-filters on semihoops. Finally, using SM-filter, we characterize two kinds of state semihoops.

Hilbert algebras are important tools for certain investigations in algebraic logic since they can be considered as fragments of any propositional logic containing a logical connective implication and the constant 1 which is considered as the logical value “true” and as a generalization of this was defined the notion of g-Hilbert algebra. In this paper, we investigate the relationship between g-Hilbert algebras, gi-algebras, implication gruopoid and BE-algebras. In fact, we show that every g-Hilbert algebra is a self distributive BE-algebras and conversely. We show cannot remove the condition self distributivity. Therefore we show that any self distributive commutative BE-algebras is a gi-algebra and any gi-algebra is strong and transitive if and only if it is a commutative BE-algebra. We prove that the MV -algebra is equivalent to the bounded commutative BE-algebra.

We prove that a closed d 0 -measure on a d 0 -algebra can be decomposed into the sum of a Lyapunov d 0 -measure and an anti-Lyapunov d 0 -measure. [ABSTRACT FROM AUTHOR]

In this paper we define the monadic pseudo BE-algebras and investigate their properties. We prove that the existential and universal quantifiers of a monadic pseudo BE-algebra form a residuated pair. Special properties are studied for the particular case of monadic bounded commutative pseudo BE-algebras. Monadic classes of pseudo BE-algebras are investigated and it is proved that the quantifiers on bounded commutative pseudo BE-algebras are also quantifiers on the corresponding pseudo MV-algebras. The monadic deductive systems and monadic congruences of monadic pseudo BE-algebras are defined and their properties are studied. It is proved that, in the case of a monadic distributive commutative pseudo BE-algebra there is a one-to-one correspondence between monadic congruences and monadic deductive systems, and the monadic quotient pseudo BE-algebra algebra is also defined. [ABSTRACT FROM AUTHOR]

Knowing the applications of logical algebras in various fields, such as artificial intelligence or coding theory, in this paper, we study some properties of a special class of such algebras, namely finite Wajsberg algebras. For this purpose, we give a representation theorem for finite Wajsberg algebras and give a formula for the number of non-isomorphic Wajsberg algebras of order n; also we give the total number of finite Wajsberg algebras of order n. Since a big value of n involves a lot of computations, as examples, we present and describe all finite Wajsberg algebras of order n ≤ 9 . [ABSTRACT FROM AUTHOR]

The theory of fuzzy filters in pseudo-BE algebras is developed. Various characterizations of fuzzy filters are given. It is proved that the set of all fuzzy filters of a pseudo-BE algebra is a complete lattice. Some characterizations of Noetherian pseudo-BE algebras by fuzzy filters are obtained. Finally, fuzzy commutative filters are defined and studied. Moreover, the homomorphic properties of fuzzy (commutative) filters are provided. [ABSTRACT FROM AUTHOR]

In this paper, the notion of a η-Smarandache pseudo fuzzy ideal of a pseudo BH- algebra is introduced. Some propositions and examples are investigated to study properties of this notion and to determine the relationships between it and a η-Smarandache pseudo ideal of a pseudo BH-algebra [ABSTRACT FROM AUTHOR]

We provide some properties of maximal Bp-subalgebras of B-algebras. In particular, we show that for each prime p, a finite B-algebra has a maximal Bp-subalgebra. We also show that for a finite B-algebra of order prm, where (p, m) = 1, any two maximal Bp-subalgebras are conjugate and the number of maximal Bp-subalgebras is kp + 1 for some k ∈ 𝕑+. [ABSTRACT FROM AUTHOR]

The notions of (∈ , ∈ ∨ q) -cubic p- (a- and q-) ideals of BCI-algebras are introduced and some related properties are investigated. Several characterization for these generalized (∈ , ∈ ∨ q) -cubic ideals are defined and relationship between (∈ , ∈ ∨ q) -cubic p-ideals, (∈ , ∈ ∨ q) -cubic q-deals and (∈ , ∈ ∨ q) -cubic a-ideals of BCI-algebras are discussed. [ABSTRACT FROM AUTHOR]

In this paper, the notions of cubic intuitionistic KU-subalgebras and KU-ideals of a KU-algebras are introduced, and several properties are investigated. Characterizations of cubic intuitionistic KU-subalgebras and KU-ideals of KU-algebras are considered. Relations between a cubic intuitionistic KU-subalgebra and a cubic intuitionistic KU-ideal are given. [ABSTRACT FROM AUTHOR]

In the paper [P.K. Maji, R. Biswas, A.R. Roy, Fuzzy soft sets, J. Fuzzy Math. 9 (3) (2001) 589–602], Maji et al. introduced the concept of fuzzy soft sets as a generalization of the standard soft sets, and presented an application of fuzzy soft sets in a decision making problem. The aim of this paper is to apply fuzzy soft set for dealing with several kinds of theories in hyper BCK-algebras. The notions of fuzzy soft hyper BCK-ideal, fuzzy soft weak hyper BCK-ideal, fuzzy soft s-weak hyper BCK-ideal and fuzzy soft strong hyper BCK-ideal are introduced, and related properties and relations are investigated. [ABSTRACT FROM AUTHOR]

Lattice implication algebra is a new logical algebraic system which is established by combining lattice and implication algebra. As a generalization of intuitionistic fuzzy set, neutrosophic set is introduced which deals with indeterminate membership in addition to degree of membership and degree of non-membership. In this article, the notion of neutrosophic set theory is applied to lattice implication algebras. The concept of implicative neutrosophic LI-ideals of a lattice implication algebra is introduced, and several properties are investigated. Relationship between a neutrosophic LI-ideal and an implicative neutrosophic LI-ideal is discussed, and conditions for a neutrosophic LI-ideal to be an implicative neutrosophic LI-ideal are provided. Characterizations of an implicative neutrosophic LI-ideal are considered, and the extension property of an implicative neutrosophic LI-ideal is studied. [ABSTRACT FROM AUTHOR]

In this paper, the notion of a QI-algebra is introduced which is a generalization of a BI-algebra and there are studied its properties. We considered ideals, congruence kernels in a QI-algebra and characterized congruence kernels whenever a QI-algebra is right distributive.

In this paper, n-ary block codes for KU-algebras and their interconnecting properties in terms of block codes have been given. A construction of n-ary block codes for a given KU-algebra is shown, and it is proved that for each n-ary block code K we can associate a KU-algebra X, such that the constructed n-ary block codes generated by X, i.e. K x , contain the code K as a subset. We have proved that for every n-ary block-code K, there exists a KU-valued function on a KU-algebra which determines K. It is also shown that the KU-algebras associated with an n-ary block code are not unique up to isomorphism. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate the concept of partially ordered B-algebras (or B po -algebras) and we provide some related properties. A B po -algebra is an algebra (X ; ∗ , ⪯ , 0) such that (X ; ∗ , 0) is a B-algebra and ⪯ is a partial order on X such that ⪯ is compatible on X, that is, property (M) holds: (M) x ⪯ y implies z ∗ (0 ∗ x) ⪯ z ∗ (0 ∗ y) and x ∗ (0 ∗ z) ⪯ y ∗ (0 ∗ z) for all x , y , z ∈ X . We also introduce partially ordered quotient B-algebra via B po -ideal. [ABSTRACT FROM AUTHOR]

The notion of implicative UP-filters is introduced, and several properties are investigated. The relation between UP-filters and implicative UP-filters is considered. Conditions for UP-filter to be implicative UP-filter are given, and characterizations of (implicative) UP-filters are discussed. An extension property for implicative UP-filter is established. [ABSTRACT FROM AUTHOR]

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.