Minimal relative generating sets of some partial transformation semigroups

Relative ranks of some partial transformation semigroups

“…In [19], we obtained the necessary and sufficient conditions for any nonempty subset U of K n,r (P K n,r ) to be a (minimal) relative generating set of T n,r (P T n,r ) modulo S n for 1 ≤ r ≤ n − 1 . Then we concluded the same result in [2,17] that rerank(T n,r : S n ) = p r (n) and, we obtained the new result…”

“…We also assume that α ∈ D P r is in the above tabular form unless stated otherwise. First, recall Proposition 1 and Lemma 2 in [19] : Next we state and prove the following similar lemma which will be used throughout this paper:…”

“…We also assume that α ∈ D P r is in the above tabular form unless stated otherwise. First, recall Proposition 1 and Lemma 2 in [19] :…”

Quasi-idempotent ranks of the proper ideals in finite symmetric inverse semigroups

“…There are studies of various ranks of semigroups, such as idempotent-rank, (m, r) rank, nilpotent rank, etc., as well as of minimal generating sets of elements of a given kind (see, for example, [2,3,6,8,13,17,18]).…”