The monoid of ordered partitions of a natural number
An operation is introduced making \(M\), the set of all sequences of nonnegative integers into a monoid shown to be isomorphic to the monoid of all partial, finite-to-one, order-preserving transformations \(f\) of the positive integers \(\mathbb{N}\), where \(\text{dom }f\) is an initial segment of \(\mathbb{N}\). A presentation is found for a submonoid \(S_n\) of \(M\) the elements of which are ordered partitions of a natural number \(n\), the posets of principal one-sided ideals of \(S_n\) are described, that for left ideals being the more complex: an algorithm is constructed for the Hasse diagram of the poset of left principal ideals of \(S_n\).
monoid of partial finite-to-one order-preserving transformations
scientific article; zbMATH DE number 895849