Certain ranks of some ideals in symmetric inverse semigroups contains S_n or A_n

Leyla Bugay


Let I_n, S_n and A_n be the symmetric inverse semigroup, the symmetric group and the alternating group on X_n={1,…,n}, for n≥2, respectively. Also let I_(n,r) be the subsemigroup consists of all partial injective maps with height less than or equal to r for 1≤r≤n-1, and let SI_(n,r)=I_(n,r)∪S_n and AI_(n,r)=I_(n,r)∪A_n. A non-idempotent element whose square is an idempotent is called a quasi-idempotent. In this paper we obtain the rank and the quasi-idempotent rank of SI_(n,r) (of AI_(n,r)). Also we obtain the relative rank and the relative quasi-idempotent rank of SI_(n,r) modulo S_n  (of AI_(n,r) modulo A_n). 

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