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

Abstract

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).

Keywords

• Symmetric inverse semigroup,quasi-idempotent,rank

Simetrik inverse yarıgrubun S_n veya A_n i içeren bazı ideallerinin rankları

Abstract

n≥2 için I_n, S_n ve A_n, sırasıyla, X_n={1,…,n} üzerindeki simetrik inverse yarıgrup, simetrik grup ve alterne grup olsun. Ayrıca, 1≤r≤n-1 için I_(n,r), yüksekliği en fazla r olan tüm kısmi bire-bir dönüşümlerden oluşan altyarıgrup, SI_(n,r)=I_(n,r)∪S_n ve AI_(n,r)=I_(n,r)∪A_n olsun. Karesi idempotent olan fakat kendisi idempotent olmayan bir elemana quasi-idempotent denir. Bu calışmada SI_(n,r) (AI_(n,r)) nin rankını elde ettik. Ayrıca, modulo S_n e göre SI_(n,r) nin (modulo A_n e göre AI_(n,r) nin) ilişkili rankını ve quasi-ilişkili rankını elde ettik.

Keywords

• Simetrik inverse yarıgrup,quasi-idempotent,rank

References

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