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.
World Drug Report 2000
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