Weak Isometries in Partially Ordered Groups

In this paper the author gives necessary and sufficient conditions under which to a stable weak isometry f in a directed group G there exists a direct decomposition G = A × B of G such that f(x) = x(A) − x(B) for each x ∈ G. Further, some results on weak isometries in partially ordered groups are established. Isometries in an abelian lattice ordered group (l-group) have been introduced and investigated by Swamy [16], [17]. Jakub́ık [4] proved that for every stable isometry f in an l-group G there exists a direct decomposition G = A × B of G such that f(x) = x(A) − x(B) for each x ∈ G. Isometries in non-abelian l-groups were also studied in [2] and [5]. Weak isometries in l-groups were introduced by Jakub́ık [6]. Rach̊unek [14] generalized the notion of the isometry for any partially ordered group (po-group). Isometries and weak isometries in some types of pogroups have been investigated in [7], [8], [9], [10], [12], [13], [14]. In [11] it was proved that each stable weak isometry in a directed group is an involutory group automorphism (hence each weak isometry in a directed group is an isometry). First we recall some notions and notations used in the paper. Let G be a po-group. The group operation will be written additively. We denote G = {x ∈ G;x ≥ 0}. If a, b are elements of G, then we denote by U(a, b) and L(a, b) the set of all upper bounds and the set of all lower bounds of the set {a, b} in G, respectively. If for a, b ∈ G there exists the least upper bound (greatest lower bound) of the set {a, b} in G, then it will be denoted by a ∨ b (a ∧ b). For each a ∈ G, |a| = U(a,−a). A partially ordered semigroup (po-semigroup) P with a neutral element is said to be the direct product of its po-subsemigroups P1 and P2 (notation: P = P1×P2) if the following conditions are fulfilled: (1) If a ∈ P1, b ∈ P2, then a+ b = b+ a. (2) Each element c ∈ P can be uniquely represented in the form c = c1 + c2 where c1 ∈ P1, c2 ∈ P2. (3) If g, h ∈ P , g = g1 + g2, h = h1 + h2 where g1, h1 ∈ P1, g2, h2 ∈ P2, then g ≤ h if and only if g1 ≤ h1, g2 ≤ h2. Received September 16, 1993; revised February 24, 1994. 1980 Mathematics Subject Classification (1991 Revision). Primary 06F15.