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Cores of Bol loops and symmetric groupoids.
Authors: A. Vanžurová
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Abstract
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, SID, or its medial subvariety, 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 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].E-mail:
Fulltext
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Contents
- Valentin Danilovich Belousov (1925-1988). The issue is a tribute in honor of his 80`th birthday.
- Pairwise orthogonality of n-ary operations.
- On some quasi-identities in finite quasigroups.
- On a small quasi-compactness.
- On the Riemann extension of the G?del space-time metric.
- Some n-ary analogs of the notion of a normalizer of an n-ary subgroup in a group.
- Loops of Bol-Moufang type with a subgroup of index two.
- On identities of Bol-Moufang type.
- A loop transversal in a sharply 2-transitive permutation loop.
- Multiobjective Games and Determining Pareto-Nash Equilibria.
- On Bruck-Belousov Problem.
- On Commutativity and Mediality of Polyagroup Cross Isomorphs.
- Cores of Bol loops and symmetric groupoids.
