Every t-Irreducible Partial Order is a Suborder of a t + Irreducible Partial Order

Abstract

The dimension of a partial order ( X , ≤) is the least integer t for which there exist linear extensions X 1 , X 2 ,…, X t , of X so that x i , ≤ x 2 in X if and only if x t , ≤ x 2 , in X , for each i = 1,2,…, t . For an integer t ≥ 2, a partial order is said to be t -irreducible if it has dimension t and every proper nonempty subpartial order has dimension less than t . We answer a natural question concerning dimension by proving that for each t ≥ 2, every t -irreducible partial order is a subpartial order of a t + 1-irreducible partial order.