A generalization of Moufang and Steiner loops. (English) Zbl 1058.20057

Kinyon, Michael K.; Kunen, Kenneth; Phillips, J. D.

Algebra Univers. 48, No. 1, 81-101 (2002).

A loop is called flexible if it satisfies the identity \(x(yx)=(xy)x\). Denote by \(L(x)\) or \(R(x)\) the so called left or right translation by \(x\), i.e. \(yL(x)=xy\), \(yR(x)=yx\). A loop is ARIF if it is flexible and satisfies the following identities \[ R(x)R(yxy)=R(xyx)R(y),\qquad L(x)L(yxy)=L(xyx)L(y). \] Every ARIF loop of odd order is Moufang whereas non-group Steiner loops are RIF and not Moufang. The main result is a generalization of Moufang’s theorem to ARIF loops: Every ARIF loop is diassociative (i.e. every two-generated subloop is a group).

Reviewer: Ivan Chajda (Olomouc)

MSC:

20N05

Loops, quasigroups

Keywords:

Moufang loops; Steiner loops; RIF loops; ARIF loops; diassociative loops

Software:

OTTER