On identities of Bol-Moufang type

A. Pavlů ; A. Vanžurová

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2005), pp. 88-100.

Résumé

(Left) Bol loops are usually introduced as loops in which (left) Bol condition is satisfied, and the existence of the two-sided inverse of any element as well as the left inverse property are deduced. It appears that some of the assumptions on the structure are superflous and can be omitted, or modified. Also, Bol loops can be presented in various settings as far as the family of operation symbols is concerned. First we give a short survey on main known results on identities of Bol-Moufang type in quasigroups, written in a unified notation, and try to employ only multiplication and left division for the equational theory of left Bol loops. Then we propose a rather non-traditional concept of the variety of left Bol loops in type $(2,1,0)$, with operation symbols $(\cdot ,{}^{-1},e)$ and with five-element defining set of identities, namely $xe=ex=x$, $(x^{-1}{)}^{-1}=x$, $x^{-1}(xy)=y$, $x(y(xz))=(x(yx))z$.

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