Geodesic
On the endomorphism semigroup of an ordered set
Glasgow mathematical journal, Tome 37 (1995) no. 2, pp. 173-178

Voir la notice de l'article provenant de la source Cambridge University Press

M. E. Adams and Matthew Gould [1] have obtained a remarkable classification of ordered sets P for which the monoid End P of endomorphisms (i.e. isotone maps) is regular, in the sense that for every f є End P there exists g є End P such that fgf = f. They show that the class of such ordered sets consists precisely of(a) all antichains;(b) all quasi-complete chains;(c) all complete bipartite ordered sets (i.e. given non-zero cardinals α β an ordered set Kα,β of height 1 having α minimal elements and β maximal elements, every minimal element being less than every maximal element);(d) for a non-zero cardinal α the lattice Mα consisting of a smallest element 0, a biggest element 1, and α atoms;(e) for non-zero cardinals α, β the ordered set Nα,β of height 1 having α minimal elements and β maximal elements in which there is a unique minimal element α0 below all maximal elements and a unique maximal element β0 above all minimal elements (and no further ordering);(f) the six-element crown C6 with Hasse diagramA similar characterisation, which coincides with the above for sets of height at most 2 but differs for chains, was obtained by A. Ya. Aizenshtat [2]. 
Blyth, T. S. On the endomorphism semigroup of an ordered set. Glasgow mathematical journal, Tome 37 (1995) no. 2, pp. 173-178. doi: 10.1017/S0017089500031074
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