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Ordered vector valued double sequence spaces

100%

Dey R. , Das N. R. , Tripathy B. C.

Fasciculi Mathematici

2015

tom Nr 55

29--34

In this paper we have introduced an order relation on convergent double sequences and have constructed an ordered vector space, Riesz space, order complete vector space in case of double sequences. We have verified the Archimedean property.

Regions-based two dimensional continua: The Euclidean case

86%

Hellman G. , Shapiro S.

Logic and Logical Philosophy

2015

tom 24

nr 4 Mereology and Beyond

499-534

We extend the work presented in [7, 8] to a regions-based, two-dimensional, Euclidean theory. The goal is to recover the classical continuum on a point-free basis. We first derive the Archimedean property for a class of readily postulated orientations of certain special regions, “generalized quadrilaterals” (intended as parallelograms), by which we cover the entire space. Then we generalize this to arbitrary orientations, and then establishing an isomorphism between the space and the usual point-based R × R. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause” (to the effect that “these are the only ways of generating regions”), and we have no axiom of induction other than ordinary numerical (mathematical) induction. Finally, having explicitly defined ‘point’ and ‘line’, we will derive the characteristic Parallel’s Postulate (Playfair axiom) from regions-based axioms, and point the way toward deriving key Euclidean metrical properties.