Symmetric 2-Structures, a Classification

Karzel, Helmut; Kosiorek, Jarosław; Matraś, Andrzej

doi:10.1007/s00025-013-0349-z

Cited by 3 publications

(5 citation statements)

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“…This paper is a continuation of our investigations on symmetric 2-structures [5]. This is a generalization of double symmetric 2-structures (cf.…”

Section: Introduction and Notationsmentioning

confidence: 83%

“…In [5] we proved that any two parallel points a and b have exactly one midpoint (cf. [5,Theorem 3.15]). …”

Section: Theorem 11 ∀A B ∈ K: A(b) ∈ K and A(b) = A • B • A Hencementioning

confidence: 97%

“…In [5] we presented a classification of symmetric 2-structures based on the cardinality of the set (p⊥K) := {L ∈ K | p ∈ L ∧ L⊥K} with p ∈ K of all chains passing through the point p and which are orthogonal to the chain K. The symmetric 2-structures split into the three classes (cf. [5,Theorem 3.10]):…”

Section: Theorem 11 ∀A B ∈ K: A(b) ∈ K and A(b) = A • B • A Hencementioning

confidence: 99%

“…[4]) which are closely related with sharply 2-transitive permutation groups [1,6]. We will use the same definitions and notations as in [5] and Σ := (P, G 1 , G 2 , K) will denote a symmetric 2-structure. We recall that for any point p ∈ P and any i ∈ {1, 2} [p] i denotes the generator of G i passing through the point p and that two points a, b ∈ P are called parallel if there is an i ∈ {1, 2} such that [a] i = [b] i and not parallel otherwise.…”

Section: Introduction and Notationsmentioning

confidence: 99%

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Point Symmetric 2-Structures

Karzel¹,

Kosiorek

Matraś

2011

Results. Math.

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Abstract. We show that every symmetric 2-structure (P, G1, G2, K) of the class (III) [cf. Karzel H et al. (Result. Math., submitted)] is point symmetric, i.e. any two orthogonal chains A, B ∈ K intersect in exactly one point and that any two points a, b ∈ P have exactly one midpoint m := a * b (with m(a) = b where m is the unique symmetry in the point m).Therefore the pair (P, P ) is an invariant regular involution set and the loop derivation in a point o ∈ P gives a K-loop (P, +) uniquely 2-divisible.Mathematics Subject Classification (2010). 51A99.

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“…This paper is a continuation of our investigations on symmetric 2-structures [5]. This is a generalization of double symmetric 2-structures (cf.…”

Section: Introduction and Notationsmentioning

confidence: 83%

“…In [5] we proved that any two parallel points a and b have exactly one midpoint (cf. [5,Theorem 3.15]). …”

Section: Theorem 11 ∀A B ∈ K: A(b) ∈ K and A(b) = A • B • A Hencementioning

confidence: 97%

Section: Theorem 11 ∀A B ∈ K: A(b) ∈ K and A(b) = A • B • A Hencementioning

confidence: 99%

Section: Introduction and Notationsmentioning

confidence: 99%

See 2 more Smart Citations

Point Symmetric 2-Structures

Karzel¹,

Kosiorek

Matraś

2011

Results. Math.

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“…In [6] we introduced a classification of symmetric 2-structures Σ depending whether for K ∈ K and p ∈ K the cardinality |(p ⊥ K) K | is > 1 (class I), is = 0 (class II) or is = 1 (class III). In [7] we showed if Σ is of class III then any two orthogonal chains intersect (in exactly one point) and called Σ point symmetric.…”

Section: Chain Structuresmentioning

confidence: 99%

A Representation of a Point Symmetric 2-Structure by a Quasi-Domain

Karzel¹,

Kosiorek

Matraś

2013

Results. Math.

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Abstract. In a symmetric 2-structure Σ = (P, G1, G2, K) we fix a chain E ∈ K and define on E two binary operations "+" and "·". Then (E, +) is a K-loop and for E * := E\{o}, (E * , ·) is a Bol loop. If Σ is even point symmetric then (E, +, ·) is a quasidomain and one has the set Af f (E, +,Af f (E, +, ·) one can reproduce via a "chain derivation" the point symmetric 2-structure Σ.Mathematics Subject Classification (1991). 51A99.

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Helmut Karzel (1928–2021)

et al. 2022

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in Burgwedel, Germany, at the age of 93. He will always be in our best memories as teacher, friend and valued member of the worldwide community of mathematicians. Our deepest sympathy goes out to his three children Dörte, Herbert, Barnim and to Gisela, his partner over the last years.Helmut Karzel was born on January 15, 1928, in Schöneck in Westpreußen, Poland. He spent his youth in Posen, where he attended school from 1934 to 1945. During the last year of World War II he had to serve as flak helper. At the end of the war he was advised to go west. So, in 1945 he reached the German town of Magdeburg, where an uncle lived. There, in 1946, he took the "Abitur" at the Otto von Guericke Schule. His mother and two of his three siblings had to flee from Posen and all of them met again in Magdeburg. 0123456789().: V,-volH. Havlicek et al. J. Geom. By crossing several occupation zone boundaries (at night), Karzel reached Freiburg, Germany, in 1947. He received the admission to study mathematics and physics at the University of Freiburg. Prior to his studies, he was obliged to help for eight weeks in clearing up the rubble of the bomb attacks. Among Karzel's teachers in Freiburg was Emanuel Sperner, who became Professor at the University of Bonn in 1950. Karzel moved to Bonn and obtained his doctorate under Sperner's supervision in 1951. That year he also received the Hausdorff Memorial Prize, which is awarded annually by the University of Bonn for the best dissertation in mathematics of the past academic year.

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Symmetric 2-Structures, a Classification

Abstract: Abstract. We classify symmetric 2-structures

Cited by 3 publications

References 5 publications

Point Symmetric 2-Structures

Point Symmetric 2-Structures

A Representation of a Point Symmetric 2-Structure by a Quasi-Domain

Helmut Karzel (1928–2021)

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