The Minimal Size of Infinite Maximal Antichains in Direct Products of Partial Orders

For a partial order P having infinite antichains by a(P) we denote the minimal cardinality of an infinite maximal antichain in P and investigate how does this cardinal invariant of posets behave in finite products. In particular we show that min{a(P), p(sqP)} ≤ a(P) ≤ a(P), for all n ∈ N, where p(sqP) is the minimal size of a centered family without a lower bound in the separative quotient of the poset P, or p(sqP) = ∞, if there is no such family. So we have a(P × P) = a(P) whenever p(sqP) ≥ a(P) and we show that, in addition, this equality holds for all infinite Boolean algebras of size ≤ ω1 (without zero), all reversed trees, all atomic posets and, in particular, for all posets of the form ⟨C,⊂⟩, where C is a family of nonempty closed sets in a compact T1-space containing all singletons. As a by-product we obtain the following combinatorial statement: If X is an infinite set and {Ai × Bi : i ∈ I} an infinite partition of the square X, then at least one of the families {Ai : i ∈ I} and {Bi : i ∈ I} contains an infinite partition ofX . 2010 Mathematics Subject Classification: 06A06, 06E10, 03E05.