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Commutative loops of exponent 3 with x.(x.y)2=y2

Armanious M.

Demonstratio Mathematica

2002

Vol. 35, nr 3

469-475

It is well known that the class of Hall triple systems [5], Steiner triple systems in which each triangle generates an affine plane over GF(3), corresponds to the class of commutative Moufang loops of exponent 3 [6]. In this paper, we extend the class of algebras to the class of all commutative loops of exponent 3 satisfying the identity x.(x.y)2=y2, corresponding to the class of all Steiner triple systems. Such a commutative loop of exponent 3 with x . (x o y)2 = y2 is polynomially equivalent to a squag.