Equality Algebras Research Articles

Constructing Symmetric Equality Algebras

In this paper, we introduce the notion of strong fuzzy filter on hyper equality algebras and investigate some equivalence definitions of it. Then by using this notion we constructed a symmetric equality algebra and define a special form of classes. By using these, we define the concept of a fuzzy hyper congruence relation on hyper equality algebra and we prove that the quotient is made by it is an equality algebra. Also, by using a fuzzy equivalence relation on hyper equality, we introduce a fuzzy hyper congruence relation and prove that this fuzzy hyper congruence is regular and finally we prove that the quotient structure that is made by it is a symmetric hyper equality algebra.

On a new sum of equality algebras

This paper explores the construction of ordinal sums in the class of equality algebras. Although the ordinal sum of linear equality algebras remains linear, this property might not hold for representable equality algebras in general. We present (prime) deductive systems for the ordinal sum of each family of equality algebras, which help us find a necessary and sufficient condition under which an ordinal sum of equality algebras is representable.

The relationship between EQ algebras and equality algebras

It is proved that every involutive equivalential equality algebra (E, ∧, ∼, 1), is an involutive residualted lattice EQ-algebra, which operation ⊗ is defined by x ⊗ y = (x → y 0 ) 0 . Moreover, it is showen that by an involutive residualted lattice EQ-algebra we have an involutive equivalential equality algebra

Roughness of filters in equality algebras

Rough is an excellent mathematical tool for the analysis of a vague description of actions in decision problems. Now, in this paper by considering the notion of an equality algebra, the notion of the lower and the upper approximations are introduced and some properties of them are given. Moreover, it is proved that the lower and the upper approximations are an interior operator and a closure operator, respectively. Also, using D-lower and D-upper approximation, conditions for a nonempty subset to be definable are provided and investigated that under which condition D-lower and D-upper approximation can be filter.

On co-annihilators in equality algebras

On co-annihilators in equality algebras

Positive implicative equality algebras and equality algebras with some types

The notion of a positive implicative equality algebras are defined, and related properties are studied. Characterizations of a positive implicative equality algebra is investigated. Conditions for an equality algebra to be positive implicative are provided. Equality algebra with some types is considered, and several properties are investigated. Using equality algebra with some types, we characterize a commutative equality algebra and a positive implicative algebra.

Derivations of equality algebras

In this paper, we introduced the concept of derivation on equality algebra \(E\) by using the notions of inner and outer derivations. Then, we investigated some properties of (inner, outer) derivation and we introduced some suitable conditions that they help us to define a derivation on \(E\). We introduced kernel and fixed point sets of derivation on \(E\) and prove that under which condition they are filters of \(E\). Finally, we prove that the equivalence relations on \((E,\rightsquigarrow ,1)\) coincide with the equivalence relations on \(E\) with derivation \( d \).

Graphs Based on Equality Algebras

In this paper, we introduce new kinds of graphs based on equality algebras. First of all, by using the meet operation we define the notion of zero divisors on equality algebra and study related properties. Then we introduce a meet graph on equality algebra by using zero divisors. In addition, we investigate some conditions for a meet graph to be a complete, connected and a star graph. Also, by using the notion of filters in equality algebras, we define an equivalence relation on equality algebras and then by using the equivalence classes we introduce two kinds of graphs on them. Finally, conditions for a graph based on these classes to be connected, or bipartite, or complete, or planar or an outer-planar graph are provided.

Quasi-Polyadic Algebras and Their Dual Position

Infinite-dimensional quasi-polyadic algebras and cylindric algebras are the best-known algebraizations of first-order logic. Quasi-polyadic equality algebras (QPEAs), due to Paul R. Halmos, are special polyadic algebras; nevertheless, for a long time, these algebras have been considered only as a version of cylindric algebras (CAs), because their behavior seemed to be very similar to that of CAs. However, in researching QPEAs, over time it turned out that many properties of QPEA are different from those of cylindric algebras. The reason is the presence of the polyadic operator, the transposition pij (missing in CAs). The author states and documents that QPEAs have a dual position in algebraic logic. In many respects, this class is like CAs, following the cylindric paradigm, but, in other respects, as an effect of the polyadic paradigm (of the transposition), QPEAs are different from CAs. The paper is a survey on quasi-polyadic algebras with particular regard to the foregoing documentation, and it fills an existing gap in the literature.

Right and Left Mappings in Equality Algebras

The notion of (right) left mapping on equality algebras is introduced, and related properties are investigated. In order for the kernel of (right) left mapping to be filter, we investigate what conditions are required. Relations between left mapping and →-endomorphism are investigated. Using left mapping and →-endomorphism, a characterization of positive implicative equality algebra is established. By using the notion of left mapping, we define →-endomorphism and prove that the set of all →-endomorphisms on equality algebra is a commutative semigroup with zero element. Also, we show that the set of all right mappings on positive implicative equality algebra makes a dual BCK-algebra.

Radical of filters on equality algebra

In this paper, by using the concept of maximal filter of equality algebra, we introduce radical of equality algebra. Then some equivalence definitions of it and some related properties are investigated. Then by using this notion, we introduce the concept of semi-maximal filter and prime-like filter on equality algebras and the relation between them and other filters of equality algebra are investigated. Finally, by using the notion of prime-like filters, we introduce a topology on equality algebra.

Tense operators on frameable equality algebras

In this paper, we introduce the frameable equality algebras and use the concept of tense operators on them to define tense equality algebras. We investigate some algebraic properties of tense equality algebras and prove the representation theory for strict strong tense equality algebras. Then, we introduce the notions of (prime) tense deductive systems and tense congruences and obtain some structural theorems.

Implicative equality algebras and annihilators in equality algebras

In this paper, the concept of an implicative equality algebra is defined and related properties are investigated. Characterizations of an implicative equality algebra and the relation among implicative and positive implicative equality algebras are discussed. Also, we define the notion of annihilator of equality algebras and investigate some traits of it and prove that annihilator of any nonempty set of equality algebras is a deductive system. Moreover, by using this notion, we define the operation and prove that for any commutative equality algebra is a bounded implicative BCK-algebra.

Filter theory on hyper equality algebras

Filter theory on hyper equality algebras

Tense like equality algebras

In this paper, first we define the notion of involutive operator on bounded involutive equality algebras and by using it, we introduce a new class of equality algebras that we called it a tense like equality algebra. Then we investigate some properties of tense like equality algebra. For two involutive bounded equality algebras and an equality homomorphism between them, we prove that the tense like equality algebra structure can be transfer by this equality homomorphism. Specially, by using a bounded involutive equality algebra and quotient structure of it, we construct a quotient tense like equality algebra. Finally, we investigate the relation between tense like equality algebras and tense MV-algebras.

An overview of hyper logical algebras

Hyper logical algebras were first studied in 2000 by Borzooei et al. They applied the concept of hyperstructures to one of the logical algebraic structures known as the BCK-algebra, and introduced two generalizations of them called the hyper BCK-algebra and hyper K-algebra. Then many researchers in this field continued their research and used hyperstructures on other logical algebras and introduced the concepts of hyper residuated lattices, hyper BL-algebras, hyper MV-algebras, hyper EQ-algebras, hyper BE-algebras, hyper equality algebras, hyper hoops and etc. Moreover, they defined some new notions such as different kinds of hyper ideals, hyper filters and hyper congruence relations on these structures and studied some properties, the relation among them and the quotient structure. Now, in this paper, we review the definitions of all those hyper logical algebras and investigate relations among them.

MBJ-neutrosophic filters of equality algebras

In this paper, we introduce the notion of MBJneutrosophic sub-algebra and MBJ-neutrosophic filter on equality algebras and investigate some equivalence definitions, properties and the relation between them. Also, by using the notion of MBJ-neutrosophic filter, we introduce a congruence relation on equality algebra and show that the quotient is an equality algebra.

Hyper equality ideals: Basic properties

In this paper, the concept of (strong) hyper equality ideals in bounded hyper equality algebras are introduced and several properties and related results are given. Also, the properties of hyper equality ideals of the direct product of bounded hyper equality algebras are investigated; we prove that any (strong) hyper equality ideal of the direct product of hyper equality algebras is representable with respect to the product of (strong) hyper equality ideals of any direct component. In the sequel, we investigate the relationships between hyper equality ideals and hyper deductive systems in good bounded hyper equality algebras. Furthermore, we show how one can construct a hyper congruence relation via a strong hyper equality ideal so that the congruence classes form a hyper equality algebra.

Fuzzy weak hyper deductive systems of hyper equality algebras and their measures

Fuzzy weak hyper deductive systems of hyper equality algebras and their measures

Results on hoops

In this paper, by considering the notion of hoop, were introduced by Bosbach in [7, 8] under the name of complementary semigroups, we show that there are relations among hoops and some of other logical algebras such as residuated lattices, MT L-algebras, BL-algebras, MV-algebras, BCK-algebras, equality algebras, EQ-algebras, R0-algebras, Hilbert algebras, Heyting algebras, Hertz algebras, lattice implication algebras and fuzzy implication algebras. The aim of this paper is to find that under what conditions hoops are equivalent to these logical algebras.