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Large rectangular semigroups in Stone-Cech compactifications

Authors:	Neil Hindman Dona Strauss Yevhen Zelenyuk

Institution:	Department of Mathematics, Howard University, Washington, DC 20059 ; Department of Pure Mathematics, University of Hull, Hull HU6 7RX, United Kingdom ; Faculty of Cybernetics, Kyiv Taras Shevchenko University, Volodymyrska Street 64, 01033 Kyiv, Ukraine

Abstract:	We show that large rectangular semigroups can be found in certain Stone-Cech 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 $\beta\mathbb{N}$ .

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