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In this paper, we introduce the notion of a (strong) hyper RL-ideal in hyper residuated lattices and give some properties and characterizations of them. Next, we characterize the (strong) hyper RL-ideals generated by a subset and give some characterizations of the lattice of these hyper RL-ideals. Particularly, we prove that this lattice is algebraic and compact elements are finitely generated hyper RL-ideals, and obtain some isomorphism theorems. Finally, we introduce the notion of nodal hyper RL-ideals in a hyper residuated lattice and investigate their properties. We prove that the set of nodal hyper RL-ideals is a complete Brouwerian lattice and under suitable operations is a Heyting algebra.

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.

This article has been retracted. A retraction notice can be found at https://doi.org/10.3233/JIFS-219433. [ABSTRACT FROM AUTHOR]

One motivation to study obstinate ideals in any algebra of logic is that the induced quotient algebra by these ideals is the two-element Boolean algebra. In this paper, we introduce two types of obstinate ideals in HvMV-algebras; obstinate HvMV-ideals and obstinate weak HvMV-ideals. Giving several theorems and examples we characterize these HvMV-ideals. For example, we prove that an HvMV-ideal (if exists) must be maximal, and any HvMV-algebra with odd number of elements does not contatin an obstinate HvMV-ideal. Also, we characterize these HvMV-ideals in finite HvMV-algebras with at most six elements; we investigate that which subsets can be an obstinate (weak) HvMV-ideal. In the sequel, we investigate the relationships between obstinate (weak) HvMV-ideals, and Boolean and prime HvMV-ideals. Finally, we prove that in a commutative HvMV-algebra, the quotient HvMV-algebra induced by an obstinate weak HvMV-ideal must be a two-elements Boolean algebra. [ABSTRACT FROM AUTHOR]

Based on Pawlak’s rough set theory, we study and investigate the roughness in non-commutative residuated lattices, which are generalizations of non-commutative fuzzy structures such as MV-algebras and BL-algebras. We give many theorems and examples to describe the rough approximations. Also, to investigate the properties of roughness of subsets (and of course filters) more closely, we consider some different kinds of filters such as Boolean filters and prime filters. Especially, we prove that with respect to some certain filters, the obtained approximations form a Boolean algebra or a pseudo MTL-algebra. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study HvMV-algebras from fuzzy set theory point of view. Some types of fuzzy ideals are introduced and many properties, characterizations and related results are given. Particularly, the fuzzy weak HvMV-ideal generated by a fuzzy subset is characterized. In the sequel, some kinds of fuzzy congruences are introduced and characterizations of them are obtained. Finally, some homomorphism theorems are stated and proved. [ABSTRACT FROM AUTHOR]