On the ranks of certain semigroups of order-preserving transformations
If \(A\) is a subset of a semigroup \(S\), we denote by \(\langle A\rangle\) the subsemigroup generated by \(A\). The rank of a finite semigroup \(S\) is defined by \(\text{rank }S=\min\{| A|: A \subseteq S, \langle A\rangle = S\}\) and if it is generated by its set \(E\) of idempotents, its idempotent rank is defined by \(\text{idrank }S = \min\{| A|: A \subseteq E, \langle A\rangle = S\}\). Now let \(X_ n = \{1,2,\dots,n\}\), let \({\mathcal P\mathcal O}_ n\) denote the semigroup of all partial order preserving transformations on \(X_ n\), denote by \({\mathcal O}_ n\) the subsemigroup of \({\mathcal P\mathcal O}_ n\) consisting of all those transformations whose domains are \(X\) and let \({\mathcal S\mathcal P\mathcal O}_ n = {\mathcal P\mathcal O}_ n\setminus{\mathcal O}_ n\). The authors obtain a number of results about the ranks and idempotent ranks of these semigroups. For example, they show that if \(n \geq 2\), then \(\text{rank }{\mathcal O}_ n = n\) and \(\text{idrank }{\mathcal O}_ n = 2n-2\). They show further that \(\text{rank }{\mathcal P\mathcal O}_ n=2n-1\), \(\text{idrank }{\mathcal P\mathcal O_ n} = 3n-2\) and \(|{\mathcal P\mathcal O}_ n| + 1\) is the coefficient of \(x^ n\) in the Maclaurin expansion of the function \((1+x)^ n(1-x)^{- n}\). Finally, they show that \(\text{rank }{\mathcal S\mathcal P\mathcal O}_ n = 2n - 2\). Since \({\mathcal S\mathcal P\mathcal O}_ n\) is not generated by idempotents, the question of its idempotent rank does not arise. It should be mentioned that the manuscript somehow became incorrectly paginated in the process of getting into print. Pages 275 and 276 were interchanged.
partial order preserving transformations
scientific article; zbMATH DE number 126128