On bounded residuated ℓEQ-algebras

Abstract

An EQ-algebra has three basic binary operations (meet, multiplication and a fuzzy equality) and a top element. An ℓEQ-algebra is a lattice-ordered EQ-algebra satisfying the substitution property of the join operation. In this article, we study bounded residuated ℓEQ-algebras (BR-ℓEQ-algebras for short). We introduce a subvariety RL-EQ-algebras of BR-ℓEQ-algebras, and prove that the categories of RL-EQ-algebras and residuated lattices are categorical isomorphic. We also prove that RL-EQ-algebras are precisely the BR-ℓEQ-algebras that can be reconstructed from residuated lattices. We further show the existence of a closure operator on the poset of all BR-ℓEQ-algebras with the same lattice and multiplication reduct, the existence of the maximum element in the poset. Then we introduce filters in BR-ℓEQ-algebras and give a lattice isomorphism between the filter lattice and the congruence lattice. Finally, we prove that the category of residuated lattices is isomorphic to a reflective subcategory of BR-ℓEQ-algebras.