Glasnik Matematicki, Vol. 52, No. 2 (2017), 247-256.

CUBIC STRUCTURE

Vladimir Volenec, Zdenka Kolar-Begović and Ružica Kolar-Šuper

Department of Mathematics, University of Zagreb, Bijenička cesta 30, HR-10 000 Zagreb, Croatia
e-mail: volenec@math.hr

Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, HR-31 000 Osijek, Croatia
e-mail: zkolar@mathos.hr

Faculty of Education, University of Osijek, Cara Hadrijana 10, HR-31 000 Osijek, Croatia
e-mail: rkolar@foozos.hr

Abstract.   In this paper we examine the relationships between cubic structures, totally symmetric medial quasigroups, and commutative groups. We prove that the existence of a cubic structure on the given set is equivalent to the existence of a totally symmetric medial quasigroup on this set, and it is equivalent to the existence of a commutative group on this set. We give also some interesting geometric examples of cubic structures. By means of these examples, each theorem that can be proved for an abstract cubic structure has a number of geometric consequences. In the final part of the paper, we prove also some simple properties of abstract cubic structures.

2010 Mathematics Subject Classification.   20N05.

Key words and phrases.   TSM-quasigroup, commutative group, ternary relation, cubic structure.

Full text (PDF) (free access)

DOI: 10.3336/gm.52.2.05

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