Cores of Bol loops and symmetric groupoids

A. Vanžurová

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2005), pp. 153-164

Résumé

The notion of a core was originally invented by R. H. Bruck for Moufang loops, [3], and the construction was generalized by V. D. Belousov for quasigroups in [2] (we will discuss 1-cores here). It is well known that cores of left Bol loops, particularly cores of Moufang loops, or groups, are left distributive, left symmetric, and idempotent, [2]. Among others, our aim is to clarify the relationship between cores and the variety of left symmetric left distributive idempotet groupoids, $\underline{SID}$, or its medial subvariety, $\underline{SIE}$, respectively. The class of cores of left Bol loops is not closed under subalgebras, therefore is no variety (even no quasivariety), and we can ask what variety is generated by cores: the class of left Bol loop cores (even the class of group cores) generates the variety of left distributive left symmetric idempotent groupoids, while cores of abelian groups generate the variety of idempotent left symmetric medial groupoids. It seems that the variety $\underline{SID}$ of left distributive left symmetric idempotent groupoids (“symmetric groupoids”) aroused attention especially in connection with symmetric spaces in 70' and 80' [15, 16, 18, 19] and the interest continues. Recently, it was treated in [8, 26, 27], and also in [29], from the view-point of hypersubstitutions. The right symmetric idempotent and medial case was investigated e.g. in [1, 21–24].