Substitution and atomic extension on greedy posets

摘要

In this paper, we study the operations of substitution and atomic extension on greedy posets. For the substitution operation, ifP=(P1, x, P2)is a greedy poset, thenP1andP2are greedy posets, the converse being false. However, for the atomic extension,P=P1(x, P2)is a greedy poset if and only ifP1andP2are greedy posets. We prove also that the class of greedy semi-partitive lattices is the smallest one containingMn(n≥2),B3and closed by atomic extension. The classCnof greedy posets with jump numbernis infinite. However, we show thatCncan be obtained, in a very simple way, from a subclassDnof finite cardinal ity. We constructDnforn=1,