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A classification of maximal inverse subsemigroups of the finite symmetric inverse semigroups. (English) Zbl 0943.20064

The symbol \(I_n\) denotes the symmetric inverse semigroup on the set \(X_n=\{1,2,\dots,n\}\) and \(S_n\) denotes the symmetric group on \(X_n\). For \(1\leq r\leq n-1\), let \(V(n,r)=\{\alpha\in I_n:|\text{im }\alpha|\leq r\}\). In the first of two main results, the author shows that if \(S\) is a maximal inverse subsemigroup of \(I_n\) then either \(S=V(n,n-2)\cup S_n\) or \(S=V(n,n-1)\cup G\) where \(G\) is a maximal subgroup of \(S_n\). In the second of the two main results, the author determines the maximal inverse subsemigroups of the inverse semigroup \(V(n,r)\).

MSC:

20M18 Inverse semigroups
20M20 Semigroups of transformations, relations, partitions, etc.
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