The endomorphism spectrum of an ordered set

Abstract

Theendomorphism spectrum of an ordered setP, spec(P)={|f(P)|:f ∈ End(P)} andspectrum number, sp(P)=max(spec(P)\{|P|}) are introduced. It is shown that |P|>(1/2)n(n − 1)n − 1 implies spec(P) = {1, 2, ...,n} and that if a projective plane of ordern exists, then there is an ordered setP of size 2n2+2n+2 with spec(P)={1, 2, ..., 2n+2, 2n+4}. Lettingh(n)=max{|P|: sp(P)⩽n}, it follows thatc1n2⩽h(n)⩽c2nn+1 for somec1 andc2. The lower bound disproves the conjecture thath(n)⩽2n. It is shown that if |P| − 1 ∈ spec(P) thenP has a retract of size |P| − 1 but that for all κ there is a bipartite ordered set with spec(P) = {|P| − 2, |P| − 4, ...} which has no proper retract of size⩾|P| − κ. The case of reflexive graphs is also treated.