  We establish new results concerning endomorphisms of a finite chain if the
cardinality of the image of such endomorphism is no more than some fixed number
k. The semiring of all such endomorphisms can be seen as a k - simplex whose
vertices are the constant endomorphisms. We explore the properties of these k -
simplices and find some results for arbitrary k. For k = 1 and 2 we give a full
description of simplices called strings and triangles, respectively.
