ON THE m-POTENT RANKS OF CERTAIN SEMIGROUPS OF ORIENTATION PRESERVING TRANSFORMATIONS

Abstract

It is known that the ranks of the semigroups <TEX>$\mathcal{SOP}_n$</TEX>, <TEX>$\mathcal{SPOP}_n$</TEX> and <TEX>$\mathcal{SSPOP}_n$</TEX> (the semigroups of orientation preserving singular self-maps, partial and strictly partial transformations on <TEX>$X_n={1,2,{\ldots},n}$</TEX>, respectively) are n, 2n and n + 1, respectively. The idempotent rank, defined as the smallest number of idempotent generating set, of <TEX>$\mathcal{SOP}_n$</TEX> and <TEX>$\mathcal{SSPOP}_n$</TEX> are the same value as the rank, respectively. Idempotent can be seen as a special case (with m = 1) of m-potent. In this paper, we investigate the m-potent ranks, defined as the smallest number of m-potent generating set, of the semigroups <TEX>$\mathcal{SOP}_n$</TEX>, <TEX>$\mathcal{SPOP}_n$</TEX> and <TEX>$\mathcal{SSPOP}_n$</TEX>. Firstly, we characterize the structure of the minimal generating sets of <TEX>$\mathcal{SOP}_n$</TEX>. As applications, we obtain that the number of distinct minimal generating sets is <TEX>$(n-1)^nn!$</TEX>. Secondly, we show that, for <TEX>$1{\leq}m{\leq}n-1$</TEX>, the m-potent ranks of the semigroups <TEX>$\mathcal{SOP}_n$</TEX> and <TEX>$\mathcal{SPOP}_n$</TEX> are also n and 2n, respectively. Finally, we find that the 2-potent rank of <TEX>$\mathcal{SSPOP}_n$</TEX> is n + 1.