The reticulation of a hoop Cheptea
In this paper we study the hoops, ordered structures introduced by B. Bosbach under the name of complemented semigroups. We built a reticulation of a hoop, which is a bounded distributive lattice such that there exists a homeomorphism between the prime spectrum of the hoop and the prime spectrum of the lattice. We find an axiomatic definition of the reticulation and we prove the existence of it. We prove the unicity of the reticulation modulo a lattices isomorphism. We built a covariant functor between the category of bounded distributive lattices and the category of bounded hoops, which allows us to obtain new properties from the properties of lattices.
Reticulation of a 0-distributive Lattice
A congruence relation θ on a 0-distributive lattice is defined such that the quotient lattice L/θ is a distributive lattice and the prime spectrum of L and of L/θ are homeomorphic. Also it is proved that the minimal prime spectrum (maximal spectrum) of L is homeomorphic with the minimal prime spectrum (maximal spectrum) of L/θ.
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 algebra A is a pair(L(A), λ) consisting of a bounded distributive latticeL(A) and a surjectionλ : 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 ofA. This construction allows many properties to be transferred between L(A) andA. 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