The generators of the Temperley-Lieb algebra generate a monoid with an appealing geometric representation. It has been much studied, notably by Louis Kau¤man. Borisavljević, Došen and Petrić gave a complete proof of its abstract presentation by generators and relations, and suggested the name 'Kau¤man monoid'. We bring the theory of semigroups to the study of a certain …nite homomorphic image of the Kau¤man monoid. We show the homomorphic image (the Jones monoid ) to be a combinatorial and regular *-semigroup with linearly ordered ideals. The Kau¤man monoid is explicitly described in terms of the Jones monoid and a purely combinatorial numerical function. We use this to describe the ideal structure of the Kau¤man monoid and two other of its homomorphic images.