On the endomorphism semigroup of an ordered set

T. S. Blyth
1995 Glasgow Mathematical Journal  
have obtained a remarkable classification of those ordered sets P for which the monoid End P of endomorphisms (i.e. isotone maps) is regular, in the sense that for every / e End P there exists g e 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 a, /3 an ordered set K a p of height 1 having a minimal elements and /3 maximal
more » ... , every minimal element being less than every maximal element); (d) for a non-zero cardinal a the lattice M a consisting of a smallest element 0, a biggest element 1, and a atoms; (e) for non-zero cardinals a,/3 the ordered set N Qj3 of height 1 having a minimal elements and )3 maximal elements in which there is a unique minimal element a 0 below all maximal elements and a unique maximal element /3 0 above all minimal elements (and no further ordering); (f) the six-element crown C 6 with Hasse diagram Proof. =$>: Suppose that End P is regular and principally ordered. Then, by [1], P tNATO CRG 910765 is gratefully acknowledged.
doi:10.1017/s0017089500031074 fatcat:ccywiq7mwvaitkvj4urs53kwo4