Unities and Negation

Abstract

We define a ‘unity’ as a preordered set with a certain separation property, imitating the preorder induced on the irreducible elements of a finite lattice. As proved in [C], any finite lattice L has a representation L ≃ 2 ^I as the set of all maps from an appropriate preordered structure I, which we call the negation of the unity of irreducibles of L, into a small unity that we call 2. In this paper, we characterize unities of irreducibles of lattices, show how arbitrary unities can be constructed from such unities of irreducibles, and discuss possible definitions of maps between unities. In this way we lay some groundwork necessary for a categorical study of unities, of the negation operator, and of functors between the category u of unities, the category P of partially ordered sets, and the category L of lattices.