On the Idempotent Ranks of Certain Semigroups of Order-Preserving Transformations

G.U. Garba

Abstract: The ranks of the semigroups $O_{n}$, $PO_{n}$ and $SPO_{n}$ (the semigroups of order-preserving singular selfmaps, partial and strictly partial transformations on $X_{n}=\{1,...,n\}$ respectively), and the idempotent ranks of $O_{n}$ and $PO_{n}$ were studied by Gomes and Howie [2]. In this paper we generalize their results in line with Howie and McFadden [7], by considering the semigroups $L(n,r)$, $M(n,r)$ and $N(n,r)$, where, for $2\le r\le n-2$, $L(n,r)=\{\alpha\in O_{n}:|\IM\alpha|\le r\}$, $M(n,r)=\{\alpha\in PO_{n}:|\IM\alpha|\le r\}$ and $N(n,r)=\{\alpha\in SPO_{n}:|\IM\alpha|\le r\}$.

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