A Half 3-Moufang Quadrangle is Moufang

Abstract

Recently, we showed that any $3$-Moufang generalized quadrangle is automatically a Moufang quadrangle. In another recent paper, Katrin Tent \cite{Ten:04a} borrowed an argument of the second author to show that the half Moufang condition implies the Moufang condition for generalized quadrangles. In the present paper we show that this argument can be used to further weaken the hypotheses: we define the half $3$-Moufang condition as a kind of greatest common divisor of the $3$-Moufang condition and the half Moufang condition and show that it implies the Moufang condition.