V-PERSPECTIVES, DIFFERENCES, PSEUDO-NATURAL NUMBER SYSTEMS AND PARTIAL ORDERS
A. Mani · University of Zagreb University Computing Centre (SRCE) · 2002
Abstract. In this paper, we generalise the notion of partial well-orderability and consider its relation to partial dierence operations possi-bly denable. Results on these and generalised PWO{posets with systems of invariants for V{PWO posets are also formulated. These are relevant in partial algebras with dierences and pseudonatural number systems for very generalised abstract model theory in particular. 1. Notations and terminology For convenience the basic notations and terminology are presented below. Set will mean a set in ZFC unless stated otherwise. A subset T of a poset S is a {subset i fx; 8 y 2 S (x y _ x k y)g T and 8x 2 S, 9 y 2 T y x. A minimal subset T is a {subset which does not properly include any {subsets i.e. fx; 8 y 2 S (x y _ x k y)g = T and 8x 2 S 9 y 2 T y x. A poset S = hS;; (2)i is well founded i each nonempty subset has at least one minimal element. A linear order is a PO which satises 8x8 y x = y_x < y_y < x. A well ordered set X is a linearly ordered set for which every nonempty set Y X has a least element, w.r.t. <. A poset S = hS;i is partially well ordered (PWO) i every subset of S has a nite {subset (but not necessarily a minimal subset) i for every in nite sequence (xn) in S there exists i; j with i < j, xi xj. All PWO{posets are well founded but not conversely and the structure is so total that every innite PWO poset S contains a chain C satisfying card(C) = card(S). All posets contain at least one {subset but this is not so