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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 $$\odot $$ -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.

Inspired by the analysis of soft sets and fuzzy sets, in this paper, we study the relevance and interrelationship of the equivalence and congruence relation of lattice ordered fuzzy soft groups(l-FSGs). Moreover, we analyze some of the innate algebraic results accrued out of it theoretically. We construct l-FSG congruence class and l-FSG quotient set on a group X. We analyze the relationship between l-FSG congruence relation and l-FSG normal subgroups. In particular, we prove that in a group X, l-FSG normal subgroups form a modular lattice.

In this paper, we focus on investigating some types of (skew) filters such as weak (positive) implicative (skew) filters on pseudo residuated skew lattices and obtain some properties of these (skew) filters. The relationship between these (skew) filters and other types of (skew) filters are discussed. It is shown that any weak positive implicative filter is a weak implicative and strong filter if it is a normal filter. Moreover, we characterize some types of filters by the properties of corresponding quotient algebras. Some equivalent characterizations for some filters are obtained too. Finally, a new characterization of these filters in a pseudo residuated skew lattice by a diagram is given.

The notions of $$(\in ,\in \vee q)$$-cubic p- (a- and q-) ideals of BCI-algebras are introduced and some related properties are investigated. Several characterization for these generalized $$(\in ,\in \vee q)$$-cubic ideals are defined and relationship between $$(\in ,\in \vee q)$$-cubic p-ideals, $$(\in ,\in \vee q)$$-cubic q-deals and $$(\in ,\in \vee q)$$-cubic a-ideals of BCI-algebras are discussed.

In the online published article, the third author’s “A. Borumand Saeid” affiliation is updated as following “Dept. of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran”.

In this paper the notion of fuzzy BE-algebras are introduced. We state and prove some theorems in fuzzy BE-algebras and level subalgebras. Some characterizations of fuzzy subalgebras of BE-algebras are given. Using fuzzy subalgebras, characterizations of Artinian and Noetherian BE-algebras are also established. Finally the notion of fuzzy topological BE-algebras is introduced. We state and prove the Foster’s results on homomorphic images and inverse images in fuzzy topological BE-algebras.