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In this paper, by using the notion of fuzzy points and equality algebras, the notions of fuzzy point equality algebra, equality-subalgebra, and ideal were established. Some characterizations of fuzzy subalgebras were provided by using such concepts. We defined the concepts of (∈, ∈) and (∈, ∈ ∨ q)-fuzzy ideals of equality algebras, discussed some properties, and found some equivalent definitions of them. In addition, we investigated the relation between different kinds of (α, β)-fuzzy subalgebras and (α, β)-fuzzy ideals on equality algebras. Also, by using the notion of (∈, ∈)-fuzzy ideal, we defined two equivalence relations on equality algebras and we introduced an order on classes of X, and we proved that the set of all classes of X by these order is a poset. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to introduce the notion of stabilizers in L-algebras and develop stabilizer theory in L-algebras. In this paper, we introduced the notions of left and right stabilizers and investigated some related properties of them. Then, we discussed the relations among stabilizers, ideal and co-annihilators. Also, we obtained that the set of all ideals of a CKL-algebra forms a relative pseudo-complemented lattice. In addition, we proved that right stabilizers in CKL-algebra are ideals. Then by using the right stabilizers we produced a basis for a topology on L-algebra. We showed that the generated topology by this basis is Baire, connected, locally connected and separable and we investigated the other properties of this topology. [ABSTRACT FROM AUTHOR]

The notion of a (limited) commutative T&F-ideal in BCK-algebras and BCIalgebras is introduced, and their properties are investigated. A relationship between a T&Fideal and a commutative T&F-ideal in BCK-algebras and BCI-algebras is established, and examples to show that any T&F-ideal may not be commutative are given. Proper conditions for a T&F-ideal to be commutative are provided. Using a commutative ideal of a BCKalgebra and a BCI-algebra, a commutative T&F-ideal is established. The closed T&F-ideal in a BCI-algebra is introduced, and a condition for a closed T&F-ideal to be commutative is discussed. Characterization of a commutative T&F-ideal in a BCI-algebra is considered. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to introduce analogously modal operators on L-algebras and study their properties. To begin with, we introduce the notion of modal operators on L-algebras and investigate some important properties of this operator. In order for the kernel of modal operator to be ideal, we investigate what conditions are required. Relations between modal operator and endomorphism of L-algebras are investigated. Also, we define the concept of positive L-algebra and some characterizations of positive L-algebra are established. Finally, we introduce a map ka and show that ka is a modal operator and we prove that the set of all ka on a positive L-algebra makes a dual BCK-algebra. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of co-annihilator on a lattice equality algebra E and investigate some basic properties of them. Specially, by using the generated filters in E, we prove that the set of all filters of E forms a bounded distributive pseudo-complemented lattice, which pseudo-complemented of any filter is co-annihilator of it. Finally, we construct a Boolean algebra by the set of all co-annihilators of E. [ABSTRACT FROM AUTHOR]

Here, we define (semi)topological L-algebras and some related results are approved. Then we deduce conditions that mention an L-algebra to be a semi-topological or a topological L-algebra and we check some attributes of them. Chiefly, we display in an L-algebra L, if (L, ↠, τ ) is a semi-topological L-algebra and {1} is an open set or L is bounded and satisfies the double negation property, then (L, τ ) is a topological L-algebra. Finally, we construct a discrete topology on a quotient L-algebra, under suitable conditions. Also, different kinds of topology such as T0 and Hausdorff are investigated. [ABSTRACT FROM AUTHOR]

In this paper, we apply the notion of n-fold filters to the EQ-algebras and introduce the concepts of n-fold pseudo implicative, n-fold implicative, n-fold obstinate, n-fold fantastic prefilters and filters on an EQ-algebra ε. Then we investigate some properties and relations among them. We prove that the quotient algebra ε/F modulo an 1-fold pseudo implicative filter of an EQ-algebra ε is a good EQ-algebra and the quotient algebra E/F modulo an 1-fold fantastic filter of a good EQ-algebra E is an IEQ-algebra. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we studied the category of EQ-algebras and showed that it is complete, but it is not cocomplete, in general. We proved that multiplicatively relative EQ-algebras have coequlizers and we calculated coproduct and pushout in a special case. Also, we constructed a free EQ-algebra on a singleton. [ABSTRACT FROM AUTHOR]

The notions of (implicative) soju filters in a hoop algebra are introduced, and related properties are investigated. Relations between a soju sub-hoop, a soju filter and an implicative soju filter are discussed. Conditions for a soju filter to be implicative are displayed, and characterizations of an implicative soju filters are considered. The extension property of an implicative soju filter is established. [ABSTRACT FROM AUTHOR]

In this paper, we define and characterize the notions of (implicative, maximal, prime) ideals in hoops. Then we investigate the relation between them and prove that every maximal implicative ideal of a V-hoop with double negation property is a prime one. Also, we define a congruence relation on hoops by ideals and study the quotient that is made by it. This notion helps us to show that an ideal is maximal if and only if the quotient hoop is a simple MV-algebra. Also, we investigate the relationship between ideals and filters by exploiting the set of complements. [ABSTRACT FROM AUTHOR]

Hoops are naturally ordered commutative residuated integral monoids, introduced by Bosbach in [6, 7], that BL-algebras are particular cases of hoops. Now, in this paper, we introduce the concept of (semi)topological hoop and we get some related results. Then we derive here conditions that imply a hoop to be a semitopological or a topological hoop and we study some properties of them. Specially, we show that in a hoop A, if (A; → T) is a semitopological hoop and f1g is an open set or A is bounded and satisfies the double negation property, then (A; T) is a topological hoop. Finally, we construct a discrete topology on quotient hoops, under suitable conditions. [ABSTRACT FROM AUTHOR]

We investigate topological structuers on a semi-hoop A and under conditions show that there exists a topology T on A such that (A, T) is a topological semi-hoop. We prove that for each cardinal number α, there exists a topological semi-hoop of order α. Finally, the separation axioms on topological semi-hoops are study and show that for any infinite cardinal number α there exists a Hausdorff topological semi-hoop of order α with non-trivial topology. [ABSTRACT FROM AUTHOR]

In this paper, we define the notions of a state ⊔-hoop and a state-morphism ⊔-hoop extending the language of hoop by adding a unary operator and investigate some properties of them. Also, we extend state ⊔-hoop to state hoop and study the relation between state hoops and state BL-algebras, MV-algebras and BCK-algebras. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 both complete and a tree under some conditions. [ABSTRACT FROM AUTHOR]

The aim of this paper is defining the concepts of Noetherian and Artinian hoops by using the filter of hoop in the partial order set of all the filters of hoops and inclusion relation and find some equivalent definitions for this notion. We translate some important results from theory of rings to the case of hoop and their characterizations are established. The relation between short exact sequence on Noetherian and Artinian hoop studied and by using short exact sequence we prove that the Cartesian product of two hoops is Noetherian (Artinian) if and only if each one is a Noetherian (Artinian). By using the notion of filter in hoops, we define the notion of composition series and prove any V-hoop is Noetherian and Artinian if and only if it has composition series. Finally, Chinese Remainder theorem in hoop and the relation between maximal filter and Noetherian (Artinian) hoop are investigated. [ABSTRACT FROM AUTHOR]