Left division in the free left distributive algebra on one generator (Q615890)

A left distributive algebra is a set \(L\) together with a binary operation \(\cdot\) on \(L\) satisfying the left distributive law \(a\cdot(b\cdot c)= (a\cdot b)\cdot (a\cdot c)\), \(a,b,c\in L\). The free left distributive algebra \(A\) on one generator is investigated in the paper. By using a division algorithm for elements of an extension of \(A\), new facts about left division in \(A\) are shown. In particular, the paper solves a conjecture raised by J. A. Moody in 2002.