## Relative ranks of some partial transformation semigroups

TURKISH JOURNAL OF MATHEMATICS, cilt.43, ss.2218-2225, 2019 (SCI İndekslerine Giren Dergi)  • Cilt numarası: 43 Konu: 5
• Basım Tarihi: 2019
• Doi Numarası: 10.3906/mat-1902-23
• Dergi Adı: TURKISH JOURNAL OF MATHEMATICS
• Sayfa Sayıları: ss.2218-2225

#### Özet

Let P-n, T-n, I-n, and S-n be the partial transformation semigroup, the (full) transformation semigroup, the symmetric inverse semigroup, and the symmetric group on X-n = {1, . . . , n}, respectively. For 1 <= r <= n - 1, let PKn,r be the subsemigroup consisting alpha is an element of P-n, such that vertical bar im alpha vertical bar <= r and let SPKn,r, = PKn,r \ T-n. In this paper, we first examine the subsemigroup I-n,I-r = I-n boolean OR PKn,r and we find the necessary and sufficient conditions for any nonempty subset of PKn,r to be a (minimal) relative generating set of the subsemigroup I-n,I-r modulo I-n. Then we examine the subsemigroups P I-n,I-r = SIn boolean OR PKn,r and SIn,r = SIn boolean OR SPKn,r for 1 <= r <= n - 1 where SIn = I-n \ S-n and compute their relative rank.