On annihilators in BL-algebras

Abstract

Abstract:In the paper, we introduce the notion of annihilators in BL-algebras and investigate some related properties of them. We get that the ideal lattice .I.L/; Â/ is pseudo-complemented, and for any ideal I , its pseudo-complement is the annihilator I ? of I . Also, we define the An.L/ to be the set of all annihilators of L, then we have that .An.L/I \; _ An.L/ ; ?; f0g; L/ is a Boolean algebra. In addition, we introduce the annihilators of a nonempty subset X of L with respect to an ideal I and study some properties of them. As an application, we show that if I and J are ideals in a BL-algebra L, then J ? I is the relative pseudo-complement of J with respect to I in the ideal lattice .I.L/; Â/. Moreover, we get some properties of the homomorphism image of annihilators, and also give the necessary and sufficient condition of the homomorphism image and the homomorphism pre-image of an annihilator to be an annihilator. Finally, we introduce the notion of˛-ideal and give a notation E.I /. We show that .E.I.L//;^E ; _ E ; E.0/; E.L/ is a pseudo-complemented lattice, a complete Brouwerian lattice and an algebraic lattice, when L is a BL-chain or a finite product of BL-chains.

On ideals in De Morgen residuated lattices

“…Examples of De Morgan residuated lattices are Boolean algebras, MV-algebras and BL-algebras (see [14,18]), MTL-algebras (see [7,11]) and Stonean residuated lattices (see [5]).…”

“…In 2015, D. Buşneag et al ([5]) investigated the variety of Stonean residuated lattices and they discussed it from the view of ideal theory. In 2016, Yu Xi Zou et al ( [18]) published a study on ideals and annihilator ideals in BL-algebras. In this paper, motivated by the previous research on co-annihilators in residuated lattices DOI: 10.14736/kyb-2018- (see [9,12,14]), ideals in MV-algebras and BL-algebras (see [8]), annihilator ideals in BL-algebras (see [18]) and implicative ideals in Stonean residuated lattices (see [5]), we study the notions of ideal and annihilator ideal in a new class of residuated lattices called De Morgan residuated lattices.…”

“…In 2016, Yu Xi Zou et al ( [18]) published a study on ideals and annihilator ideals in BL-algebras. In this paper, motivated by the previous research on co-annihilators in residuated lattices DOI: 10.14736/kyb-2018- (see [9,12,14]), ideals in MV-algebras and BL-algebras (see [8]), annihilator ideals in BL-algebras (see [18]) and implicative ideals in Stonean residuated lattices (see [5]), we study the notions of ideal and annihilator ideal in a new class of residuated lattices called De Morgan residuated lattices. This paper is organized as follows: in the next section we give some preliminaries including the basic definitions, some examples of residuated lattices, rules of calculus and theorems that are needed in the sequel.…”

The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices

“…In 2015, Buşneag et al [8] investigated the variety of Stonean residuated lattices from the view of ideal theory. In 2016, Zou et al [9] published a study on ideals and annihilator ideals in BL-algebras. In 2016, Paad [10] published a study on integral ideals and maximal ideals in BLalgebras.…”

“…In this paper, motivated by the previous research on ideals in MV-algebras [5,6], BL-algebras [5,9,10], Stonean residuated lattices [8] and De Morgan residuated lattices [4], also by the research (based on filter theory) on the prime, maximal spectra, the stable topology and pure filters in BL-algebras [12,14] and residuated lattices [15], we think that it will be interesting to study from ideal theory point of view the prime, maximal spectra, the stable topology and pure ideals and reticulation in varieties of residuated lattices. In the general case of residuated lattices, the development of an ideal theory can face many difficulties, for example, the concepts of prime ideals and ∩-irreducible (∩-prime) ideals are not equivalent,…”