Independent definition of reticulations on residuated lattices

A notion of reticulation which provides topological properties on algebras has introduced on commutative rings in 1980 by Simmons in [5]. For a given commutative ring A, a pair (L, λ) of a bounded distributive lattice and a mapping λ : A → L satisfying some conditions is called a reticulation on A, and the map λ gives a homeomorphism between the topological space Spec(A) consisting of prime filters of A and the topological space Spec(L) consisting of prime filters of L. The concept of reticulation are generalized to non-commutative rings, MV-algebras ([1]), BL-algebras ([3]), quantale ([2]) and so on. Since these algebras are axiomatic extensions of residuated lattices which are algebraic semantics of so-called fuzzy logic, it is natural to consider properties of reticulations on residuated lattices. In 2008, Mureşan has published a paper about reticulations on residuated lattices and she has provided an axiomatic definition of reticulations on residuated lattices, in which five conditions are needed. In this short note, we show that only two independent conditions of reticulation are enough to axiomatize reticulations on residuated lattices and also prove that reticulations on residuated lattices can be considered as homomorphisms between residuated lattices and bounded distributive lattices.