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Properties of $n$ -ary groups connected with the affine geometry are considered. Some conditions for an $n$ -ary $r s$ -group to be derived from a binary group are given. Necessary and sufficient conditions for an $n$ -ary group $< θ, b >$ -derived from an additive group of a field to be an $r s$ -group are obtained. The existence of non-commutative $n$ -ary $r s$ -groups which are not derived from any group of arity $m < n$ for every $n \geq 3$ , $r > 2$ is proved.

On the geometry of field extensions.

Hans Havlicek (1993)

Aequationes mathematicae

On the planar translation structures

Mauro Biliotti, Giuseppe Micelli (1992)

Rendiconti del Seminario Matematico della Università di Padova

One configurational characterization of Ostrom nets

Jaromír Baštinec (1991)

Mathematica Bohemica

Bz the quadrileteral condition in a given net there is meant the following implication: If $A_{1}, A_{2}, A_{3}, A - 4$ are arbitrary points, no three of them lie on the same line, with coll $(A_{i} A_{j})$ (collinearity) for any five from six couples ${i, j}$ then there follows the collinearity coll $(A_{k} A_{l})$ for the remaining couple ${k, l}$ . In the article there is proved the every net satisfying the preceding configuration condition is necessarity the Ostrom net (i.e., the net over a field). Conversely, every Ostrom net satisfies the above configuration...

One generalization of quasifield

Jarolím Bureš, Josef Klouda (1976)

Mathematica Slovaca

Places of alternative fields

Alena Vanžurová (1981)

Sborník prací Přírodovědecké fakulty University Palackého v Olomouci. Matematika

Planare Neokörper mit additiven Kürzungsregeln.

Otto Bachmann (1972)

Mathematische Zeitschrift

Produkte von Kongruenzklassengeometrien universeller Algebren.

Heinrich Werner (1971)

Mathematische Zeitschrift

Quadratische Semi-Ordnungen und Quadratische Formen.

A. Prestel (1973)

Mathematische Zeitschrift

Reflection loops of spaces with congruence and hyperbolic incidence structure

Alexander Kreuzer (2004)

Commentationes Mathematicae Universitatis Carolinae

In an absolute space $(P, 𝔏, \equiv, α)$ with congruence there are line reflections and point reflections. With the help of point reflections one can define in a natural way an addition + of points which is only associative if the product of three point reflection is a point reflection again. In general, for example for the case that $(P, 𝔏, α)$ is a linear space with hyperbolic incidence structure, the addition is not associative. $(P, +)$ is a K-loop or a Bruck loop.

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Isotopic invariants of natural planar ternary rings

Kleinsche Geometrie und Theorie der geometrischen Objekte

Moufang planes and the groups EK6 and SL2(K), K a Cayley division algebra.

Natural planar ternary rings

New closure conditions and some problems in loop theory.

On a generalization of nets

On Groupoids Having a Linear Congruence Class Geometry.

On Rusakov’s $n$ -ary $r s$ -groups

On the geometry of field extensions.

On the planar translation structures

One configurational characterization of Ostrom nets

One generalization of quasifield

Places of alternative fields

Planare Neokörper mit additiven Kürzungsregeln.

Produkte von Kongruenzklassengeometrien universeller Algebren.

Quadratische Semi-Ordnungen und Quadratische Formen.

Reflection loops of spaces with congruence and hyperbolic incidence structure

Regularity of algebras with applications to congruence class geometry

Ternary rings with a left quasi-zero belonging to translation planes

Translation structures and group partitions

Currently displaying 21 – 40 of 53