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

Abstract

For a partial order $\mathbb {P}$ having infinite antichains by $\mathfrak {a}(\mathbb {P})$ we denote the minimal cardinality of an infinite maximal antichain in $\mathbb {P}$ and investigate how does this cardinal invariant of posets behave in finite products. In particular we show that $\min \{ \mathfrak {a}(\mathbb {P}),\mathfrak {p} (\text {sq} \mathbb {P}) \} \leq \mathfrak {a} (\mathbb {P}^{n} ) \leq \mathfrak {a} (\mathbb {P})$, for all $n\in \mathbb {N}$, where $\mathfrak {p} (\text {sq} \mathbb {P})$ is the minimal size of a centered family without a lower bound in the separative quotient of the poset $\mathbb {P}$, or $\mathfrak {p} (\text {sq} \mathbb {P})=\infty$, if there is no such family. So we have $\mathfrak {a} (\mathbb {P} \times \mathbb {P})=\mathfrak {a} (\mathbb {P})$ whenever $\mathfrak {p} (\text {sq} \mathbb {P})\geq \mathfrak {a} (\mathbb {P})$ and we show that, in addition, this equality holds for all posets obtained from infinite Boolean algebras of size ≤ø ₁ by removing zero, all reversed trees, all atomic posets and, in particular, for all posets of the form $\langle \mathcal {C} ,\subset \rangle$, where $\mathcal {C}$ is a family of nonempty closed sets in a compact T ₁-space containing all singletons. As a by-product we obtain the following combinatorial statement: If X is an infinite set and {A _i ×B _i :i∈I} an infinite partition of the square X ², then at least one of the families {A _i :i∈I} and {B _i :i∈I} contains an infinite partition of X.