Free distributive groupoids (Q1262396)

Let F be an absolutely free groupoid and E a free monoid of rank 2. If r,s,u,v,w\(\in F\) are such that \(r=u.vw\) is a subterm of s and if \(e\in E\) is its address in s, then \(\bar e(s)\) is the term obtained from s after replacing r by uv.uw. All these partial transformations \(\bar e,\) \(e\in E\), generate a monoid and the main result of the paper is the following: Theorem. Let \(f,g\in S\). Then \(hf=kg\) for some \(h,k\in S\) such that \(dom(hf)=dom(f)\cap dom(g)\).