Chaines alternées. (Alternating chains) (Q1066925)

Let P be a poset and A a fixed subset of P. A chain \(C\subseteq P\) is said to be alternating if for every two distinct elements a,b of C there exist elements c,d in C between a and b, such that A separates each of the pairs a,c and b,d. It is proved that the following conditions are equivalent: (i) A is a Boolean combination of final segments (i.e., increasing subsets) of P; (ii) the lengths of alternating chains are bounded. If, moreover, P has no infinite antichain, then the above conditions are also equivalent to the following one: (iii) there is no infinite alternating chain.