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Let A be an MV-algebra. An (⊙ , ∨) -derivation on A is a map d : A → A satisfying: d (x ⊙ y) = (d (x) ⊙ y) ∨ (x ⊙ d (y)) for all x , y ∈ A . This paper initiates the study of (⊙ , ∨) -derivations on MV-algebras. Several families of (⊙ , ∨) -derivations on an MV-algebra are explicitly constructed to give realizations of the underlying lattice of an MV-algebra as lattices of (⊙ , ∨) -derivations. Furthermore, (⊙ , ∨) -derivations on a finite MV-chain are enumerated and the underlying lattice is described. [ABSTRACT FROM AUTHOR]

In a large number of multivalued logic and fuzzy logic of algebraic systems, residuated lattices play a prominent role and have considerable applications. States operators have been introduced on residuated lattices, and their properties are useful for the development of an algebraic theory of probabilistic models of those algebras. In this paper, we introduce the notion of state ideal in the framework of state residuated lattices, investigate some related properties, and provide several examples. Also, we present two types of state residuated lattices: state i-simple residuated lattices and state i-local residuated lattices, and characterize them. Moreover, the relationship between state ideals and state filters is analyzed using the set of complement elements. Furthermore, we prove that the lattice of all state ideals of a given state residuated lattice is a complete lattice. The notion of obstinate state ideals in state residuated lattices is also introduced, and several characterizations are presented. [ABSTRACT FROM AUTHOR]

In this paper, we characterize those rings A and complete Brouwerian residuated lattices L whose residuated lattice F i d (A , L) , of L-fuzzy ideals of A , forms a Boolean algebra. [ABSTRACT FROM AUTHOR]

In this paper, we define the notion of EBL-algebras, which are generalizations of BL-algebras and EMV-algebras. The notions of ideals, congruences and filters in EBL-algebras are introduced, and their mutual relationships are investigated. There is a one-to-one correspondence between the set of all ideals in an EBL-algebra and the set of all congruences on an EBL-algebra. Moreover, we give a representation theorem on EBL-algebras. Every proper EBL-algebras under some condition can be embedded into an EBL-algebras with a top element as an ideal. [ABSTRACT FROM AUTHOR]

In 1988, given a complete Brouwerian lattice L : = (L ; ∧ , ∨ ; 0 , 1) and a ring A : = (A ; + , · ; - ; 0) with unity 1, Swamy and Swamy (J Math Anal Appl 134:94–103, 1988) built a lattice structure, on the set of L-fuzzy ideals of A , and investigated some of its arithmetic properties. Since the residuation theory is richer than the lattice theory [see, Ciungu (Non-commutative multiple-valued logic algebras, Springer monographs in mathematics, Springer, Berlin, 2014), Galatos et al. (An algebraic glimpse at substructural logics, volume 151 of studies in logic and the foundations of mathematics, Elsevier, Amsterdam, 2007), Jipsen and Tsinakis (in: Martinez (ed) Ordered algebraic structures, Kluwer Academic Publisher, Dordrecht, 2002), Piciu (Algebras of fuzzy logic, Editura Universitaria Craiova, Craiova, 2007)], in this paper, we consider the notion of fuzzy ideals rather under a complete Brouwerian residuated lattice L : = (L ; ∧ , ∨ , ⊖ , ↠ , ⊸ ; 0 , 1) . A residuated lattice F i d (A , L) : = (F i d (A , L) ; ∧ , + , ⊗ , ↪ , ↬ ; χ 0 , 1 ̲) is built on the set F i d (A , L) of L-fuzzy ideals of A and it is shown that the latter is both an extension of L and the residuated lattice I d (A) : = (I d (A) ; ∩ , + , ⊙ , → , ⇝ ; { 0 } , A) on the set I d (A) of ideals of A . [ABSTRACT FROM AUTHOR]

In this paper, we define the notion of a pseudo-eBE-algebra as an extension of a pseudo-BE-algebra, and it is studied in detail. The construction of an eBE-algebra from a pseudo-eBE-algebra is given. Further, the notions of filters and ideals are considered. The classes of distributive and commutative pseudo-eBE-algebras are introduced and investigated. We prove that for a distributive pseudo-eBE-algebra filters coincide with ideals. Also, some types of filters are defined and the relationship between these is investigated. [ABSTRACT FROM AUTHOR]

In this paper, we define the notion of uniform BL-algebras and derive some conditions under which the operations of BL-algebras are uniformly continuous. Also, some properties of uniform topology are discussed. Finally, we use some types of congruence relations to construct some uniformities and analyze the relationship between these uniformities. [ABSTRACT FROM AUTHOR]

The rough approximations on a complete completely distributive lattice L based on binary relation were introduced by Zhou and Hu (Inf Sci 269:378-387, 2014), where the binary relation was defined on the set of non-zero join-irreducible elements. This paper extends Zhou and Hu's rough set model by defining new approximation operators via ideal. When I is the least ideal of L and R is a reflexive binary relation, these two approximations coincide. We present the essential properties of new approximations and also discuss how the new one relates to the old one. Finally, the topological and lattice structures of the approximations are given. [ABSTRACT FROM AUTHOR]

MV-algebras as well as orthomodular lattices can be seen as a particular case of so-called 'basic algebras' which are an alter ego of bounded lattices whose sections are equipped with fixed antitone involutions. The class of basic algebras is an ideal variety. In the paper, we give an internal characterization of congruence kernels (ideals) and find a finite basis of ideal terms, with focus on monotone and effect basic algebras. We also axiomatize basic algebras that are subdirect products of linearly ordered ones. [ABSTRACT FROM AUTHOR]