
On the rank of semigroup C F_{(n,r)}
 GONG Heyu, SHU Qin, ZHAO Ping

2024, 59(10):
122126.
doi:10.6040/j.issn.16719352.0.2023.111

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Let T_{n} be the full transformation semigroup on X_{n}={1,2,…,n}. Let 1≤r≤n, putF_{(n,r)}={α∈T_{n}:iα=i, ∠i∈{1,2,…,r}},it is obvious that the semigroup F_{(n,r)} is subsemigroup of T_{n}. In the paper, we study the core(C F_{(n,r)})=〈E(F_{(n,r)})〉 of the semigroup F_{(n,r)}, where E(F_{(n,r)})={α∈F_{(n,r)}:α^{2}=α}, by analyzing idempotents of the semigroup F_{(n,r)}, we prove that the rank and idempotent rank of semigroup C F_{(n,r)} are both equal to ((nr)(nr1))/2+r(nr)+1.