Intersections of isotone clones on a finite set

Let ≤ be a fixed order of height at least 2 on a set A (i.e., contains a chain a < b < c). It is shown that all the isotone clones preserving orders on A isomorphic to ≤ intersect in the clone K_A of trivial functions (i.e., all the projections and all the constant operations on A). It is further shown that for A finite with at least eight elements and for any six-element set there exist two orders on A such that every joint endomorphism is trivial (i.e., id_A or constants). The same is true for intersections of isotone clones. This yields that with the above restrictions there are four maximal isotone clones intersecting in K_A. Separate considerations are given on the intersections of maximal isotone clones for |A| = 3 and 4.