This paper deals with ordered patitions of a set (indexed by some integer r) considered as 'natural' extensions of subsets, mainly from the lattice theory viewpoint (secondary, ring theory aspects). The present theory is in fact a (complete) set-theoretical realization of general Post algebras. This study --which we had briefly sketched in a previous paper (Serfati . cf. Chap. G., L'alg~bre de Post des r-partitions ordonnfes d'un ensemble, 35-37) --is completely defined here and developed to a full extent.
Resum6
Cet article est consacr6 aux partitions ordonn~es d'un ensemble f2 (index~es par un entier r), considfrfes comme des extensions 'naturelles' des sous-ensembles de ~, essentiellement du point de vue des treillis (et, secondairement, de celui des anneaux). La thforie ici prfsentfe est en fait une rfalisation ensembliste compl&e des alg~bres de Post abstraites. Cette 6tude-que nous avions bri~vement esquissfe dans un prfcfdent article (Serfati [5]. cf. Chap. G., L'algbbre de Post des r-partitions ordonnfes d'un ensemble, 35-37) --est ici redffinie et developpfe d'une faqon complbte. (~) 1999 Elsevier Science B.V. All fights reserved
O. Introduction and preliminaries
We use the following notations: supremum and infimum of any two elements x and y of a lattice T will be denoted by x V y and x A y. If it exists, the supremum (resp. infimum) of any indexed family xi (i C 1) of elements of T will be denoted by
~1 Xi (resP'i~lXi) "
Universal elements (if they exist) will be denoted by 0 and 1: that is T is a (0, 1)lattice. Recall that in any (0, 1)-distributive lattice, the set of all the complemented