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.
Archimedean Residuated Lattices
Archimedean Residuated Lattices For a residuated lattice A we denote by Ds(A) the lattice of all deductive systems (congruence filters) of A. The aim of this paper is to put in evidence new characterizations for maximal and prime elements of Ds(A) and to characterize archimedean and hyperarchimedean residuated lattices; so we prove some theorems of Nachbin type for residuated lattices. These results generalize to the case of residuated lattices some results earlier obtained by Buşneag and Piciu for the case of BL-algebras.