Quasi-idempotent ranks of some permutation groups and transformation semigroups

Authors: LEYLA BUGAY

Abstract: Let $S_{n}$, $A_{n}$, $I_{n}$, $T_{n}$, and $P_{n}$ be the symmetric group, alternating group, symmetric inverse semigroup, (full) transformations semigroup, and partial transformations semigroup on $X_{n}=\{1,\dots ,n\}$, for $n\geq 2$, respectively. A non-idempotent element whose square is an idempotent in $P_{n}$ is called a quasi-idempotent. In this paper first we show that the quasi-idempotent ranks of $S_{n}$ (for $n\geq 4$) and $A_{n}$ (for $n\geq 5$) are both $3$. Then, by using the quasi-idempotent rank of $S_{n}$, we show that the quasi-idempotent ranks of $I_{n}$, $T_{n}$, and $P_{n}$ (for $n\geq 4$) are $4$, $4$, and $5$, respectively.

Keywords: Symmetric/alternating group, full/partial transformations semigroup, symmetric inverse semigroup, quasi-idempotent, rank

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