Vol. 106, No. 2, 1983

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A note on the cardinality of infinite partially ordered sets

John Norman Ginsburg

Vol. 106 (1983), No. 2, 265–270

Let P be an infinite partially ordered set with 0 and 1. A subset B of P is called a π-base for P if for every element x of P with 0 < x < 1 there exist elements b, c in B such that 0 < b x c < 1. We let π(P) denote the smallest cardinality of a π-base for P. We also let (P) = sup{π(S) : S P}. The width and depth of P are defined as usual: w(P) = sup{κ : P contains an antichain of cardinality κ}; d(P) = sup{κ : P contains a well-ordered or dually well-ordered subset of cardinality κ}. We establish the following result: Theorem. |P|≤ (P)d(P). Various corollaries are obtained which imply and extend several known results on the cardinality of partially ordered sets, for example: Corollary. (a) |P|≤ 2(P). (b) |P|≤ w(P)d(P). (c) If B is a Boolean algebra then |B|≤ 2w(B).

Mathematical Subject Classification
Primary: 06A10, 06A10
Secondary: 04A20
Received: 2 June 1981
Revised: 13 April 1982
Published: 1 June 1983
John Norman Ginsburg