An Algebraization of Linear Continuum Structures
This paper continuous the approach of developing an order-theoretic structure theory of one-dimensional continuum structures as elaborated in [Wi07] (see also [Wi83],[Wi03]). The aim is to extend the order-theoretic structure theory by a meaningful algebr
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stract. This paper continuous the approach of developing an ordertheoretic structure theory of one-dimensional continuum structures as elaborated in [Wi07] (see also [Wi83],[Wi03]). The aim is to extend the order-theoretic structure theory by a meaningful algebraization; for this, we concentrate on the real linear continuum structure with its derived concept lattice which gives rise to the so-called “real half-numbers”. The algebraization approaches an ordered algebraic structure on the set of all real half-numbers to make the continuum structure of the reals more transparent and tractable.
Contents 1. Introduction 2. Linear Continuum Structures 3. Concept Lattices Derived from Ordered Sets 4. An Algebraization by Real Half-Numbers 5. Further Research
1
Introduction
It is still an open question whether a continuum should consist of points or not. This paper is based on the view that, on the phenomenological level, a continuum does not contain points, but, on the logical level, it might be useful to construct points as limits of convergence processes. Aristotle made this already clear by analysing Zenon’s paradox of the flying arrow: it says that the flying arrow is in each moment in some place, hence does not change the place in any moment, and therefore does not move. Aristotle has convincingly analysed this paradox by making clear “that the movement is not performed in a ‘now’, but in some time; time however does not consist of nows, but of durations” ([We72], p.431). Aristotle understood time and durations as continua which means according to his continuum definition that they “are unlimitedly divisible into smaller parts” ([We72], p.431). Therefore, for Aristotle, durations do not consist of time points, but time points are only limits of durations. Aristotle’s conception of the time and space continuum yields in general that a continuum does not consist of R. Medina and S. Obiedkov (Eds.): ICFCA 2008, LNAI 4933, pp. 150–157, 2008. c Springer-Verlag Berlin Heidelberg 2008
An Algebraization of Linear Continuum Structures
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points, but has as parts only continua again whose nature is to be extensive. In contrast to that, points are in principle of different nature: they are not extensive and can only be understood as limits of extensives.
2
Linear Continuum Structures
First we recall the mathematical analysis of the one-dimensional continuum of an unlimited straight line which is based by the following definition cited from [Wi07]: Definition 1. A linear continuum structure is defined as an ordered set C := (C, ≤) satisfying the following conditions: (1) C is a -semilattice with greatest element 1 and without smallest element; (2) the ∧-irreducible elements of C form two disjoint dense chains C and C without greatest and smallest element, where c ∨ c = 1 for all c ∈ C and c ∈ C ; (3) c1 ∧ c2 = d1 ∧ d2 implies c1 = d1 , c2 = d2 for all c1 , d1 ∈ C , c2 , d2 ∈ C ; (4) there exists an antiisomorphism c → c from C onto C such that C = {1} ∪ C ∪ C ∪ {c ∧ d | c ∈ C and d ∈ C
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