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,wx,y,z,w in the fixed order can be bracketed in five ways:

(1)(xy)(zw),x((yz)w),(x(yz))w,((xy)z)w,x(y(zw)). (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,ca,a,b,c than we can order them in six ways such that bb and cc are in fixed order. Bol-Moufang identities are obtained when we take general elements a,a,b,ca,a,b,c with bb before cc (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)(ab)(ac) = a((ba)c). We can do this in (52)×6=60\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

Last revised on October 29, 2013 at 11:30:30. See the history of this page for a list of all contributions to it.