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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., Comment: 24 pages, 6 figures