Large rectangular semigroups in Stone-Cech compactifications

Abstract:

We show that large rectangular semigroups can be found in certain Stone-Čech compactifications. In particular, there are copies of the $2^{\mathfrak {c}}\times 2^{\mathfrak {c}}$ rectangular semigroup in the smallest ideal of $(\beta \mathbb {N},+)$, and so, a semigroup consisting of idempotents can be embedded in the smallest ideal of $(\beta \mathbb {N},+)$ if and only if it is a subsemigroup of the $2^{\mathfrak {c}}\times 2^{\mathfrak {c}}$ rectangular semigroup. In fact, we show that for any ordinal $\lambda$ with cardinality at most $\mathfrak {c}$, $\beta {\mathbb {N}}$ contains a semigroup of idempotents whose rectangular components are all copies of the $2^{\mathfrak {c}}\times 2^{\mathfrak {c}}$ rectangular semigroup and form a decreasing chain indexed by $\lambda +1$, with the minimum component contained in the smallest ideal of $\beta \mathbb {N}$. As a fortuitous corollary we obtain the fact that there are $\leq _{L}$-chains of idempotents of length $\mathfrak {c}$ in $\beta \mathbb {N}$. We show also that there are copies of the direct product of the $2^{\mathfrak {c}}\times 2^{\mathfrak {c}}$ rectangular semigroup with the free group on $2^{\mathfrak {c}}$ generators contained in the smallest ideal of $\beta \mathbb {N}$.