Locally maximal regular subsemibands of the finite transformation semigroups $${\mathcal {T}}(n,r)$$ T ( n , r )

Abstract

Let $${\mathcal {T}}_n$$ be the semigroup of all full transformations on the finite set $$X_n=\{1,2,\ldots , n\}$$ . For $$1 \le r \le n$$ , set $${\mathcal {T}}(n, r)=\{\alpha \in {\mathcal {T}}_n: {\mathrm {rank}}(\alpha )\le r\}$$ . In this paper we describe the locally maximal regular subsemibands of the finite transformation semigroups $${\mathcal {T}}(n,r)$$ and completely obtain their classification.