On ideals of residuated lattices

Abstract

In this paper, we first point out some mistakes in [12]. Especially the Theorem 3.9 [12] showed that: Let A be residuated lattice and ∅ ≠ X ⊆ A, then the least ideal containing X can be expressed as: 〈X〉 = {a ∈ A|a ≤ (·· · ((x₁ ⊕ x₂) ⊕ x₃) ⊕ ·· ·) ⊕ x _n , x _i  ∈ X, i = 1, 2 ·· · , n}. But we present an example to illustrate the ideal generation formula may not hold on residuated lattices. Further we give the correct ideal generation formula on residuated lattices. Moreover, we extend the concepts of annihilators and α-ideals to MTL-algebras and focus on studying the relations between them. Furthermore, we show that the set I _α  (M) of all α-ideals on a linear MTL-algebra M only contains two trivial α-ideals {0} and M. However, the authors [24] studied the structure of I _α  (M) in a linear BL-algebra M, which means some results with respect to I _α  (M) given in [24] are trivial. Unlike that, we investigate the lattice structure of I _α  (M) on general MTL-algebras.