nLab identity of Bol-Moufang type

Identities of Bol-Moufang type are 60 fourth-order identities for a binary algebraic operation which include left and right Bol identities and the three Moufang identities. The identities follow from associativity but are strictly weaker than associativity; in the case of loops some of them do imply associativity.

Four general elements $x,y,z,w$ in the fixed order can be bracketed in five ways:

$$(xy)(zw), x((yz)w), (x(yz))w, ((xy)z)w, x(y(zw)).$$

If the two of the four elements are the same (hence we have $a,a,b,c$ than we can order them in six ways such that $b$ and $c$ are in fixed order. Bol-Moufang identities are obtained when we take general elements $a,a,b,c$ with $b$ before $c$ (for definiteness) and form a fourth-order monomial in any bracketing and equate it with a monomial with a different bracketing, for example $(ab)(ac) = a((ba)c)$. We can do this in $\binom{5}{2} \times 6= 60$ ways. Requiring different sets among Bol-Moufang identities in a quasigroup or in a loop one gets some interesting classes of quasigroups and loops, for example, Bol loops, Moufang loops and some others.

The notion is introduced in

• F. Fenyves, Extra loops II. On loops with identities of Bol–Moufang type, Publ. Math. Debrecen 16 (1969), pp. 187–192

Important clarifications are introduced in

• Kenneth Kunen, Quasigroups, loops and associative laws, J. Algebra 185 (1) (1996), pp. 194–204 doi (for all identities of Bol-Moufang type); Moufang quasigroups, J. Algebra 183:1 (1996) 231–234 doi (the question of loop vs. quasigroup in presence of Moufang identities)
• A. Pavlů, A. Vanžurová, On identities of Bol-Moufang type, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2005, no. 3, 88–100 pdf

A systematic exposition is also in

• Vladimir Volenec, Grupoidi, kvazigrupe i petlje (book in Croatian), Zagreb, Školska knjiga, 1982

category: algebra