By Steinke G. F.

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10. 6 is proved in a diﬀerent way. 4]. Yet another way to prove this theorem is to use Cayley’s theorem on the number of labeled trees: If we have a tree, say Γ on the set {1, 2, . . , n + 1} of vertices, we can consider it as a rooted tree with the root (n + 1). In particular, we have a well-deﬁned notion of a distance of a vertex to the root. Now we can deﬁne the nilpotent element α from PT n as follows: If i is a vertex, let j denote the unique vertex of Γ such that (i, j) is an edge and whose distance to the root is smaller than that of i.

Im1 ][im1 +1 , im1 +2 , . . , im2 ] · · · [imk−1 +1 , imk−1 +2 , . . , imk ]. To get the above expression from the permutation i1 , i2 , . . , in of 1, 2, . . , n, it is enough to choose the ends im1 , im2 , . . , imk−1 of the ﬁrst k − 1 chains (as mk = n automatically). This can be done in n−1 k−1 diﬀerent ways. Going through all permutations we will get chain-cycle notation for all nilpotent elements of defect k. Since the order of chains in the chain-cycle notation is not important (because all chains in the chain-cycle notation commute), every nilpotent element of defect k will be counted k!

This means that (ef )−1 = f e. However, since ef is an idempotent, we also have (ef )−1 = ef . Thus f e = ef and we are done. A semigroup S is called commutative or abelian if ab = ba for all a, b ∈ S. 6 it follows that the set E(S) of idempotents of an inverse semigroup S is a commutative subsemigroup of S. In particular, E(IS n ) is a commutative semigroup of order 2n . 7 Prove that εA · εB = εA∩B for all A, B ⊂ N and use this to show that the semigroup E(IS n ) is isomorphic to the semigroup of all subsets of N with respect to the operation of the intersection of subsets.