Linear Order in Three Dimensional Euclidean and Double Elliptic Spaces

Hallett, George H.

doi:10.2307/2007257

Cited by 3 publications

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“…Then it is impossible to extend our definition of + (Definition 2) to all pairs of points of S in such a way as to preserve the associative law. 9 Proof. Suppose such an extension of Definition 2 possible in 5-it being understood of course that the iterated sums appearing in the associative law are defined by Definition 3.…”

Section: Suppose (1) Then By Definition 3 (3)mentioning

confidence: 99%

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Spherical Geometries and Multigroups

Prenowitz

1950

Can. j. math.

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1. Introduction. The notion spherical geometry is suggested by the familiar geometry of the Euclidean 2-sphere in which the role of path is played by “arc of great circle”. The first postulational treatment of the subject seems to be that of Halsted [10] for the two-dimensional case. Kline [11] under the name double elliptic geometry, gave a greatly simplified foundation for the three-dimensional case based on the primitive notions point and order. Halsted and Kline study not merely descriptive (that is positional, non-metrical) properties of figures but also introduce metrical notions by postulating or defining congruence. Kline includes a continuity postulate designed to yield real spherical geometry.

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Section: Suppose (1) Then By Definition 3 (3)mentioning

confidence: 99%

“…Compare Veblen[17, Assumption 5] ; see also Flanders[8, Axiom 05] and his reference to Hallett[9] 5. A multigroup is a system closed under an associative many-valued operation °, which contains elements x,y satisfying the relations a° xD b, y ° aD b when a,b are in the system; see[6, pp.…”

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confidence: 99%