Sums and products of ordered systems

Abstract

ordered systems are given. We shall use some of the notation and definitions of [3]. For the sake of convenience we list here some of those definitions and symbols that will be employed. By an ordered system is meant a nonempty set R of elements in which a reflexive binary relation r = r' is defined. Unless otherwise specified, an italic capital letter always will denote an ordered system in the sequel. A subsystem T of R is a subset of elements of R with the order relation in T imposed by that in R. The expressions and symbols maximal element, greatest element, ascending chain condition, isomorphic, >, and so on, will have their usual meanings (see [2], for example). The symbols V and A will be used in denoting least upper bound (l.u.b.) and greatest lower bound (g.l.b.) respectively. The symbols 0 and I will denote the bounds of bounded ordered systems. The term number will mean partially ordered set. The symbol S>R will mean that R is isomorphic to a subsystem of 5. If for each element r in R, Sr is an ordered system, the ordered sum over R of the systems Sr (denoted by ~Y+rSt) is the system P where the elements of P are the ordered pairs (r, s) with r in R and s in 5r, and (r, s) = (r', s') means that r > r' or else r = r' and s = s'. If all Sr = S, we write R o S for ^rS?. The ordered product over R of the Sr (denoted by IJa-SV) is the system P where the elements of P are the functions/ defined on R such that f(r)ESr, while /^/' means that if f(r) ?af'(r), there exists r'^r such that/(r') >f'(r'). We list several results in [3 ] that are used in proofs in this paper. [3,2.2] states that ^,rSt>R and ^,RSr>St for every element t