L'Intervalle En Theorie Des Relations; Ses Generalisations Filtre Intervallaire Et Cloture D'Une Relation

Abstract

Abstract Given a chain, or total ordering ⩽, an interval can be defined in two ways : absolute and relative. An absolute interval is a subset I of the base, such that if x, y ∈ I and x ⩽ z ⩽ y, then z ∈ I. A relative interval with bound {a, b} where a t. A key for generalization to an arbitrary relation A, is the notion of local automorphism. A bijective function f is a local automorphism of A, if Dom f and Rng f (range) are subsets of the base, and if f is an isomorphism of the restriction A/Dom f onto A/Rng f. Then a subset I of the base |A| is an absolute A-interval, if any local automorphism f of the restriction A/I is extensible by the identity map on |A|-I; i.e. the union of f and of the latter identity map is still a local automorphism of A. Any intersection of A-intervals is an A-interval. A characterization is given for the exterval (complement of an interval). A subset D of the base is a relative A-interval with bound F (subset of the base), if D is disjoint from F and is maximal, with respect to inclusion, among those sets D such that any local automorphism of A/D is extenseible by the identity map on F. Every absolute A-interval is a relative A-interval. Both notions are identical for a chain A, but already differ for a partial ordering A. Two related notions are introduced : those of finite-val and subval. The finite-val is a boolean notion : union, intersection, complement of finite-vals are finite-vals. For a circular ordering C defined from a chain A by C (x, y, z) = + iff x ⩽ y ⩽ z (mod A) or any condition obtained from this by a circular permutation; then I is a C-subval iff I is an A-absolute interval or exterval; intuitively the subval is the circular segment, or cake-portion. Every interval or exterval is a subval; every subval is a finite-val, but not conversely. An A-filter (A-ultrafilter) is defined as usually, by replacing sets by A-absolute intervals. A compactable relation is defined in two equivalent ways : (1) for every A-interval I, the complement of I is a finite union of A-intervals; (2) to each A-ultrafilter F, associate an element h (F) ∈ F; then there exist finitely many F such that the union of corresponding h (F) covers the base. Finally A-ultrafilters and usual ultrafilters are used to extend to arbitrary relations: (1) the closure of rationals by real numbers; (2) the Stone closure of a boolean lattice.