On the Existence of Nonnegativity Domains of Subsets of Groups

Abstract

The most important particular cases of the main results of this paper say that: A subset A of the group X has a nonnegativity domain if and only if A is symmetric and 2-cancellable. Moreover, A has an additive nonnegativity domain if and only if A is symmetric and perfectly cancellable. In addition, it is shown that a commutative subset of X is perfectly cancellable if and only if it is infinitely cancellable.Here, the set A is called n-cancellable for some natural n if nx = 0 implies x = 0 for all x € A. In particular, A is called infinitely cancellable if it is n-cancellable for all n. Moreover, A is called perfectly cancellable if for any additive antisymmetric subset B of A and any x € A \ (B U (-B)) we have either 0 ^ Fg(x) or 0 ^ Fg (-x), where Fg(x) is the additive hull of (B U {a:}) \ {0}.The results obtained are illustrated here only with the help of additive and multiplicative groups of complex numbers. Their applications to odd, additive and translation relations are postponed to subsequent papers. We do not compare our results with the more complicated results of P. Conrad, R. Botto Mura and A. Rhemtulla on the extensions of partial right-orders to total ones in multiplicative groups. 1991 Mathematics Subject Classification: 06P15. A few basic facts on families of sets and subsets of groups

Sums and products of intervals in ordered groups and fields

“…The main purpose of this article is to show that equations (1), (2) remain valid in more general settings. Our references to ordered structures are [4], [8], [12], [13], [18]. Now, we shall give a sort list of the necessary concepts and notations: We say that X = X( ) is a partially ordered set or a poset if X is a set and is a relation on X such that it is reflexive, symmetric and transitive.…”

Sums and products of intervals in ordered semigroups

“…Our references for ordered algebraic structure are [4], [9], [10], [11], [13]. Now we give a short list of necessary concepts: We say that X = X( ) is a partially ordered set or poset, if X is a set and is a relation on X such that it is reflexive, symmetric and transitive.…”