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Answers and Hints to Exercises Answers and Hints to Exercises 1.5.3 (a): 7, (b): 19, (c): 6, (d): 16, (e): 45. 1.5.4 8!. − n T − n PT 1.5.6 (a): (n k) for n and (n +1 k) for n, (b): k (−1)i k (n − i)n for T and k (−1)i k (n +1− i)n for PT , i=0 i n i=0 i n k − i k − n T k − i k − n PT (c): i=0( 1) i (k i) for n and i=0( 1) i (k +1 i) for n. Hint: For (b) and (c) use the inclusion–exclusion formula. 1.5.7 Hint: Use Cayley’s theorem on the number of labeled trees. 1.5.8 Hint: See [Hi1, 6.1.1(b)] 1.5.9 (a): nn − (n − 1)n for at least one fixed element and n · (n − 1)n−1 for exactly one fixed element, (b): (n +1)n − nn for at least one fixed element and nn for exactly one fixed element. 2.10.3 Hint: To each α ∈PTn associate α ∈Tn+1 as follows: α(i),i∈ dom(α); α(i)= n +1, otherwise. 2.10.9 Hint: Use Corollary 2.7.4. 2.10.10 Hint: Use Corollary 2.7.4. 2.10.18 Hint: Rewrite the signless Lah number L (n, n − k) in the form − n n−1 L (n, n k)= k k k! and use Theorem 2.5.1 and Corollary 2.8.6. 2.10.20 Hint: Use Theorem 2.5.1. 2.10.23 Hint: If α = 0, the equation α·x = 0 may have at most nn solutions, whereas the equation x · α = 0 may have at most (n +1)n−1 solutions. But nn > (n +1)n−1. 3.3.2 (a) Hint: Use Cayley’s theorem on the number of labeled trees. 3.3.3 (a) Hint: Sn is not commutative. (c) Hint: Show that one can write (i, i +1),i<n,asaproductof(1, 2) and powers of (1, 2,...,n). 277 278 ANSWERS AND HINTS TO EXERCISES (l) 4.4.10 Hint: For each a ∈ S the condition of the exercise gives elements ea (r) (l) (r) (l) (r) and ea such that ea a = aea = a. Show first that ea b = b and bea = b (l) (r) for all a, b and then that ea = ea = e is the identity element of S. 4.8.1 Hint: To prove the first equality count how many different partitions of N ∪{n +1} generate the same partition of N. To prove the second equality count the number of those partitions of N ∪{n +1} for which the block containing the element n + 1 has cardinality k +1. 4.8.3 Hint: The number of anti-chains in B(N) is smaller than the cardinality of B(N). On the other hand, each collection of elements from B(N) having the same cardinality is an anti-chain. 4.8.4 (a): 5; (b): 19; (c): 167. 4.8.7 For n>4 the function f(x)=xn/nx satisfies the condition f(2) > 1 and f(n) = 1 and has on [2,n] the unique local extremal point, namely, some local maximum. 4.8.8 (a): 1, 25, 200, 600, 600, 120; (b): 5, 300, 1, 500, 1, 200, 120; (c): 1, 155, 1, 800, 3, 900, 1, 800, 120. 4.8.9 Hint: If S is inverse, e, f ∈ S are idempotents and eS = fS, then fe = e, ef = f and thus e = f. If every principal one-sided ideal is generated by a unique idempotent and b, c areinversetoa, then for the idempotents ab, ac, ba and ca we have abS = aS = acS, Sba = Sa = Sca, which implies ab = ac, ba = ca and b = bac = c. 4.8.10 In and In−1. 4.8.11 In and In−1. 4.8.12 In. 4.8.13 (a): kn−k; (b): (k +1)n−k. 4.8.14 (a): n1n2 ···nk; (b): n1n2 ···nk. 4.8.18 (a): (n +1)n−rank(α); (b): (|dom(α)| +1)n. 4.8.19 Hint: Each such ideal contains all idempotents of rank one. If α =0 (α = 0) for every idempotent of rank one, then α =0. 5.1.5 Hint: Use the proof of Theorem 4.7.4. 5.5.1 Hint: Each maximal subgroup is an H-class, but each H-class of both Tn and ISn is at the same time an H-class of PT n. 5.5.2 (d) Hint: If b is an inverse to a and ab = ba, then e = ab is an idempotent and a is invertible in eSe. 5.5.8 (a) Hint: Every generating system of a must contain a. (b) Hint: If p1,p2,...,pk are pairwise different primes and m = p1p2 ···pk, then {m/p1, m/p2,...,m/pk} is an irreducible generating system of Zm. 5.5.9 (a) Hint: An element of type (k, m)inTn is uniquely determined by an ordered partition N = N1 ∪ N2 ∪···∪Nk+1, a collection of mappings Ni → Ni+1, i =1,...,k, and a permutation of order m on Nk+1. (b) Hint: Additionally to (a) we have a (possibly empty) block N0 = dom(α). (c) An element of type (k, m)inISn is uniquely determined by an ordered partition ANSWERS AND HINTS TO EXERCISES 279 N = N1 ∪ N2 ∪···∪Nk+1, a collection of injective mappings Ni → Ni+1, i =1,...,k, and a permutation of order m on Nk+1. 5.5.11 Hint: S =(Z3, +), T = {0}. · n−1 n(n−1) 5.5.12 n 2 + 2 +3. 5.5.13 (b) Hint: No. Consider the semigroup (N, ·) and its subsemigroups pN and pqN, where p and q are different primes. 5.5.14 Hint: Consider an R-class and an L-class of a rectangular band. 6.6.3 These are all decompositions into left resp. right cosets with respect to some subgroup. 6.6.4 No. 6.6.8 Hint: We have to count the number of collections k1 <k2 < ··· < l2 <l1 from Theorems 6.5.3 and 6.5.4. Use the fact that the number of ways to write n asasumn = n1 + ···+ nt of positive integers (here t can n−1 vary), equals 2 .ForISn all necessary collections can be obtained in the following way: write n + 1 as the sum n +1 = n1 + ···+ nt, t>1, if t is odd, delete the last summand, and then consider the collection n1 − 1, n1 + n2 − 1, n1 + n2 + n3 − 1,. For PT n all necessary collections can be obtained in the following way: write n + 1 as the sum n +1=n1 + ···+ nt, t>1, and consider the collection n1 − 1, n1 + n2 − 1, n1 + n2 + n3 − 1,.... If t is odd we can either delete the last summand or double the middle one. If t is even, we can either take this collection, or delete the last summand and double the middle one of those which are left. One has also to take into account that Δ coincides with its own transpose. 7.4.11 Hint: Follow the proof of Theorems 7.4.1 and 7.4.7. 7.5.7 Hint: Follow the proof of Corollaries 7.5.1 and 7.5.6. 7.7.2 0 and 0a, a ∈ N. 7.7.4 14. 7.7.5 {ε, 0} for ISn and PT n and {ε} for Tn. 7.7.6 Hint: Determine first all automorphism which induce the identity map- ping on the set of all idempotents of I1. 7.7.7 Hint: Go through the lists given by Theorems 7.4.1, 7.4.7, and 7.4.10. 7.7.9 Hint: Each finite semigroup has an idempotent. At the same time mapping the whole semigroup to an idempotent is always an endomorphism. 7.7.11 Each endomorphism of (N, +) has the form x → k · x forsomefixed k ∈ N. 8.2.9 Hint: Show that Nτm canbemappedtoNτn using an inner automor- phism of PT n. Analogously for ISn. 8.6.1 Hint: Consider the cyclic semigroup of type (k, 1). 280 ANSWERS AND HINTS TO EXERCISES 8.6.4 (b): For example, S is a nontrivial group, T consists of one element and ϕ is the unique mapping from S to T . 8.6.5 Hint: Take the product a1 ···alk and consider the factors a1 ···ak, ak+1 ···a2k,..., a(l−1)k+1 ···akl of this product. Show that they all belong to X. 8.6.6 Hint: Consider all possible linear combinations of the elements of T and show that they form a nilpotent subalgebra of Matn(C). 8.6.7 Hint: Prove that each maximal subsemigroup of Matn(C) is a subalge- bra and use flags of subspaces in Cn instead of partial orders. 8.6.9 Hint: Consider the subsemigroup T = In ∪{[1, 2, 3][4] ...[n]} of ISn. The maximal nilpotent subsemigroups of T , corresponding to the natural linear order and the linear order 3 < 2 < 1 < 4 < ··· <n, have different cardinalities. n ··· · | | 8.6.10 1 + k=2(m1 + m2 + + mi−1) mi, where mi = Mi for all i. 8.6.11 Hint: Use Theorem 8.4.12. 9.6.2 {(at,at+lm):t ≥ k, l ≥ 1}∪{(an,an):n ≥ 1}. 9.6.9 Write α as a product α= μν, where μ is a permutation and ν = {i1,...,im} B1 ··· Bk is an idempotent in Tn.
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