On finite algebras having a linear congruence class geometry. (English) Zbl 0549.51008
Author’s abstract: ”According to Wille a congruence class geometry arises from a (universal) algebra essentially by taking the elements of the base set of the algebra as points and the congruence classes as subspaces of the geometry. Such a geometry is called linear if every of its lines is specified uniquely by any two of its points. In theorem 3.8 all those nonsimple finite algebras (with at least one fundamental operation of arit\(y\geq 2)\) having a linear congruence class geometry are determined. In particular, it is proved that the congruence class geometry of each such algebra is affine and desarguesian.”
Reviewer: J.André
MSC:
| 51E30 | Other finite incidence structures (geometric aspects) |
| 51E99 | Finite geometry and special incidence structures |
| 51A30 | Desarguesian and Pappian geometries |
References:
| [1] | J. Andr?,?ber Parallelstrukturen. Teil I: Grundbegriffe. Math. Z.76 (1961), 85-102. · Zbl 0099.15401 · doi:10.1007/BF01210963 |
| [2] | J. Andr?,?ber Parallelstrukturen. Teil II: Translationsstrukturen. Math. Z.76 (1961), 155-163. · doi:10.1007/BF01210967 |
| [3] | M. Biliotti,Strutture di Andre ed S-Spazi con traslazioni Geom. Dedic.10 (1981), 113-128. · Zbl 0459.51002 · doi:10.1007/BF01447415 |
| [4] | P. M. Cohn,Universal Algebra. Harper and Row, New York-Evanstone-London, 1965. |
| [5] | P. Dembowski,Finite Geometries. Springer-Verlag, Berlin-Heidelberg-New York, 1968. |
| [6] | G. Gr?tzer,Universal Algebra. D. van Nostrand, Princeton, N.J., 1968. |
| [7] | M. Hall, Jr.,The Theory of Groups. Macmillan, New York, tenth printing, 1968. |
| [8] | B. Huppert,Endliche Gruppen I. Springer-Verlag, Berlin-Heidelberg-New York, 1967. · Zbl 0217.07201 |
| [9] | T.Ihringer,On Certain Linear Congruence Class Geometries. Proc. Int. Conf. Combinatorial Geometries and their Applications (Rome 1981). Annals of Discrete Mathematics (to appear). · Zbl 0527.51001 |
| [10] | T. Ihringer,On Groupoids Having a Linear Congruence Class Geometry. Math. Z.180 (1982), 395-411. · doi:10.1007/BF01214179 |
| [11] | P. Libois,Quelques espaces lin?aires. Bull. Soc. Math. Belg.16 (1964), 13-22. · Zbl 0132.40601 |
| [12] | P. P.P?lfy, Unary Polynomials in Algebras, I. Preprint. |
| [13] | A. Pasini,On the Finite Transitive Incidence Algebras. Boll. Un. Mat. Ital. (5)17-B (1980), 373-389. · Zbl 0437.20031 |
| [14] | H. Werner,Produkte von Kongruenzklassengeometrien universeller Algebren. Math. Z.1210 (1971), 111-140. · Zbl 0203.22902 · doi:10.1007/BF01113481 |
| [15] | R. Wille,Kongruenzklassengeometrien. Lecture Notes in Mathematics113, Springer-Verlag, Berlin-Heidelberg-New York, 1970. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
