Abstract: It is well-known that the (left or right) Cayley graph for a finitely generated group has 0,1,2 or infinitely many ends. In joint work over recent years, Vesna Kilibarda and I have defined the number of ends for finitely generated semigroups and monoids. For arbitrary m,n we have constructed examples of monoids T such that the left Cayley graph for T has m ends while the right Cayley graph for T has n ends. In this talk, I will present the Layer Lemma which provides a reasonably transparent and uniform construction for vast numbers of such examples. I will outline a proof of the Layer Lemma.