Characterization of the Reticulation of a Residuated Lattice

The reticulation of an algebra was first defined for commutative rings by Simmons [19] and it was extended by Belluce to non-commutative rings [3]. As for the algebras of fuzzy logics, Belluce also constructed the reticulation of an MV-algebra [2], G. Georgescu defined the reticulation of a quantale [8] and L. Leuştean made this construction for BL-algebras [13, 14]. In each of the papers cited above, although it is not explicitely defined this way, the reticulation of an algebraA is a pair (L(A), λ) consisting of a bounded distributive lattice L(A) and a surjection λ : A → L(A) such that the function given by the inverse image of λ induces (by restriction) a homeomorphism of topological spaces between the prime spectrum of L(A) and that of A. This construction allows many properties to be transferred between L(A) and A. In [15] we defined the reticulation of a residuated lattice A as a pair (L(A), λ) consisting of a bounded distributive lattice L(A) and a function λ : A → L(A) satisfying five conditions of algebraic nature, proved its