Maximal regular subsemigroups of certain semigroups of transformations (Q1601811)

\(T_n\), \(P_n\), and \(S_n\) denote, respectively, the full transformation semigroup on the set \(X_n=\{1,2,\dots,n\}\), the partial transformation semigroup on \(X_n\), and the symmetric group on \(X_n\). Let \(J_r=\{\alpha\in T_n:|X_n\alpha|=r\}\) and let \(K(n,r)=\bigcup^r_{i=1}J_i\). The author shows that a maximal regular subsemigroup of \(T_n\) is either \(K(n,n-2)\cup S_n\) or \(K(n,n-1)\cup G\) where \(G\) is a maximal subgroup of \(S_n\). He also shows that for \(2\leq r\leq n-1\), each maximal regular subsemigroup of \(K(n,r)\) is either \(K(n,r-1)\cup(J_r\setminus L_\beta)\) or \(K(n,r-1)\cup(J_r\setminus R_\alpha)\) where \(\alpha,\beta\in J_r\), \(L_\beta\) is the \(\mathcal L\)-class containing \(\beta\) and \(R_\alpha\) is the \(\mathcal R\)-class containing \(\alpha\). Next, let \(PJ_r=\{\alpha\in P_n:|X_n\alpha|=r\}\) for \(0\leq r\leq n\) and let \(PK(n,r)=\bigcup^r_{i=0}PJ_i\). He shows that a maximal regular subsemigroup of \(P_n\) is either \(PK(n,n-2)\cup J_{n-1}\cup S_n\) or \(PK(n,n-1)\cup J_{n-1}\cup S_n\) or \(PK(n,n-1)\cup G\) for some maximal subgroup \(G\) of \(S_n\). He concludes by showing that for \(1\leq r\leq n-1\), the maximal regular subsemigroups of \(PK(n,r)\) are of the form \(PK(n,r)\setminus R_\alpha\) for some \(\alpha\in PJ_r\).