On orders of two transformation semigroups of the boolean

Abstract

We consider the semigroup $\mathcal{O}(\mathcal{B}_n)$ of all order-preserving transformations $\varphi : \mathcal{B}_n \rightarrow \mathcal{B}_n$ of ordered by inclusion boolean $\mathcal{B}_n$ of $n$-element set (i.e. such transformations that $A \subseteq B$ implies $\varphi(A) \subseteq \varphi(B)$) and its subsemigroup $\mathcal{C}(\mathcal{B}_n)$ of those transformations for which $\varphi(A) \subseteq A$ for all $A \in \mathcal{B}_n$. Orders of these semigroups are calculated.