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

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 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 < ···

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. im b1 ··· bk

10.1.1 Hint: For every a ∈ M the set {ϕ(s)(a):s ∈ S} is invariant. 10.1.2 For example, the natural action of the subsemigroup S of ISn,gen- erated by the element [1, 2,...,n], on N. 10.7.1 This action is always faithful but never transitive. It is quasi-transitive unless S = Sn,orS = Tn. 10.7.2 Yes. 10.7.3 The action is faithful. It is neither transitive nor quasi-transitive un- less S = S1 = T1. 10.7.4 The action is faithful. It is not transitive unless S = S1 = T1.Itis not quasi-transitive unless S = S1 = T1,orS = IS1 = PT 1. 10.7.5 The action is faithful and quasi-transitive but never transitive. 10.7.7 Only the trivial action. 10.7.12 Only the trivial action and the actions trivially induced from Sn (i.e., where only the action of invertible elements is different from 0). Hint: If n> 1andϕ : Tn →ISm is a homomorphism, different from the ones listed above, then by Theorem 6.3.10 the homomorphism ϕ is injective on idempotents of rank (n − 1). However, such idempotents of Tn do not commute in general. 10.7.13 Only the trivial action and the actions trivially induced from Sn (i.e., where only the action of invertible elements is different from 0). Hint: See the hint to 10.7.12. 10.7.14 Hint: It is enough to consider the actions of the form ξe,H and show that for any π ∈Sn and α ∈ISn such that π|dom(α) = α we have α·x = π·x. 10.7.15 (b) Hint: Consider the action a · x = ax. ANSWERS AND HINTS TO EXERCISES 281

n 11.1.5 Hint: for S = Sn and S = Tn the subspace {(c1,...,cn) ∈ C : c1 + ···+ cn =0} is a submodule of the natural module. 11.5.1 Hint: Use 11.4.4(ii). 11.7.3 If A ⊂ N, then the simple module corresponding to A is CA = C with the action B · c = c, c ∈ C, B ⊂ N,ifA ⊂ B and B · C = 0 otherwise. Every module is a direct sum of simple modules. 11.7.4 n. 11.7.5 For example, the module V (M), where M is the trivial H(01)-module. 11.7.9 (b) Hint: A finite semigroup is nilpotent if and only if it has a zero element and this zero element is the only idempotent. 11.7.11 Hint: All elements s ∈ S\G annihilate M ⊗C V . 11.7.17 Hint: Consider for example the semigroup S = x : x2 = x3. 11.7.18 A rectangular band has two simple modules: the trivial module, and the one-dimensional module with the zero multiplication. A rectangular band is a monoid if and only if it has only one element. Hence in the case, when a rectangular band has exactly one element, and we consider it as a monoid, there is only one simple module: the trivial one. 11.7.19 (b) Hint: Let X be a submodule of V (M), which is not contained in N(M). Show first that there exists v ∈ X such that e·v = 0. Show then that e · v ∈ X is a nonzero vector in the H(e)-submodule M of V (M). Finally, use the construction of V (M) to show that such X contains V (M). 11.7.20 (a) Hint: Let X be a simple S-module. Show first that there exists e ∈E(S) such that e · X = 0, while f · X = 0 for all idempotents f ∈ SeS such that f ∈D(e). Let M = e · X and M be a simple submodule of M. Similarly to Lemma 11.4.3 show that there is a nonzero homomorphism from ∼ V (M )toX. Finally, use Exercise 11.7.19 to conclude that X = V (M ).

12.3.7 Hint: Show that any two H-cross-sections can be transferred into each other by conjugation. IO≺ 12.8.6 There are theree cross-sections of the form 3 , and an additional cross-section from Exercise 12.8.5. 12.8.8 Hint: Use the Miller-Clifford Lemma ([Hi1, Theorem 1.2.5]), which states that for any a, b ∈ S we have ab ∈R(a)∩L(b) if and only if R(b)∩L(a) contains an idempotent. 12.8.9 The R-classes. Hint: Use Exercise 12.8.8.

13.6.3 Hint: Show that ϕ :(S, ◦a) → (S, ◦b) is an isomorphism. 13.6.4 Hint: Use Exercise 13.6.3. 13.6.5 Hint: Count the number of idempotents. 13.6.6 Hint: Count the number of those elements which cannot be written as a product of other elements. 13.6.10 Hint: Use Exercise 13.6.9. 282 ANSWERS AND HINTS TO EXERCISES

p n n − k k n 13.6.12 Answer: p!. m p − m p p=0 m=1 k n m m 13.6.13 Answer: nn if k =1,nn − S(k, m) S(n − k, j) j!if m j m=1 j=1 k>1. 13.6.15 Hint: Use the fact that (·)−1 : B → B is an involution.

14.2.7 Reference: See for example [Grm, 10.5]. 14.2.10 Hint: Use induction on n. − 14.6.1 Hint: From√ the recursive√ relationf2k =3f2(k−1) f2(k−2) deduce f = √1 ((3 + 5)/2)k + ((3 − 5)/2)k , insert this formula into the sum 2k 5 from Theorem 14.3.7 and use the binomial formula. 14.6.3 (a) n(n − 1). (b) n(n − 1)/2. (c) n(2n − 1). (d) n(n +1)/2. 14.6.4 The graph of such transformation has two connected components with m and n − m vertices, respectively. Such components are in bijection with nilpotent elements from Cn and Cn−m, respectively. Using Proposi- tion 14.5.5(ii), one obtains a recursive relation for the number of elements in question, and it remains to use Exercise 14.2.7. 14.6.6 Hint: Use Exercise 14.6.5. 14.6.7 Hint: Use Exercise 14.6.5. 14.6.9 Hint: The first factor gives the number of ways to choose im(α), the second factor gives the number of partitions of N into r intervals. 14.6.15 If α ∈PFn is not an idempotent (that is, not all x ∈ im(α) are fixed points), then rank(α2) < rank(α). 14.6.16 (a) Hint: Choose maximal elements in α−1(1),. . . , α−1(k − 1). 14.6.23 Hint: Use Theorem 14.2.8(i), Proposition 14.3.1 and Stirling’s for- mula for n!. 14.6.25 Hint: Use Exercise 14.6.24. 14.6.27 Hint: (b) R(α)={β :im(β)=im(α)}. Let im(α)={a1,...,ak}, a1 < ···

[APR] R. Adin, A. Postnikov, Y. Roichman, Combinatorial Gelfand models. J. Algebra 320 (2008), no. 3, 1311–1325.

[Ai1] A. Aizenshtat, Defining relations of finite symmetric semigroups. (Russian) Mat. Sb. N.S. 45(87) (1958), 261–280.

[Ai2] A. Aizenshtat, The defining relations of the endomorphism semi- group of a finite linearly ordered set. (Russian) Sibirsk. Mat. Zh. 3 (1962), 161–169.

[Ai3] A. Aizenshtat, On the semi-simplicity of semigroups of endomor- phisms of ordered sets. (Russian) Dokl. Akad. Nauk SSSR 142 (1962), 9–11.

[Ai4] A. Aizenshtat, On homomorphisms of semigroups of endomor- phisms of ordered sets. (Russian) Leningrad. Gos. Ped. Inst. Uchen. Zap. 238 (1962), 38–48.

[AAH] G. Ayık, H. Ayık, J. Howie, On factorizations and generators in transformation semigroups. Semigroup Forum 70 (2005), no. 2, 225–237.

[BRR] D. Borwein, S. Rankin, L. Renner, Enumeration of injective par- tial transformations. Discrete Math. 73 (1989), no. 3, 291–296.

[Ca] R. Carmichael, Introduction to the theory of groups of finite order. Dover Publications, New York, 1956.

[Cat] P. Catarino, Monoids of orientation-preserving transformations of a finite chain and their presentations. Semigroups and applica- tions (St. Andrews, 1997), 39–46, World Sci. Publ., River Edge, NJ, 1998.

[CH] P. Catarino, P. Higgins, The monoid of orientation-preserving mappings on a chain. Semigroup Forum 58(2) (1999), 190–206.

[Ch1] K. Chase, Sandwich semigroups of binary relations. Discrete Math. 28(3) (1979), 231–236.

283 284 BIBLIOGRAPHY

[Ch2] K. Chase, Maximal groups in sandwich semigroups of binary re- lations. Pacific J. Math. 100(1) (1982), 43–59.

[CP1] A. Clifford, G. Preston, The algebraic theory of semigroups. Vol. I. Mathematical Surveys, No. 7. American Mathematical Society, Providence, RI, 1961.

[CP2] A. Clifford, G. Preston, The algebraic theory of semigroups. Vol. II. Mathematical Surveys, No. 7. American Mathematical Society, Providence, RI, 1967.

[CR] D. Cowan, N. Reilly, Partial cross-sections of symmetric inverse semigroups. Int. J. Algebra Comput. 5(3) (1995), 259–287.

[C-O] K. Cvetko-Vah, D. Kokol Bukovˇsek, T. Koˇsir, G. Kudryavtseva, Y. Lavrenyuk, A. Oliynyk, Semitransitive and transitive sub- semigroups of the inverse symmetric semigroups, Preprint arXiv:0709.0743.

[DF] M. Delgado, V. Fernandes, Abelian kernels of monoids of order- preserving maps and of some of its extensions. Semigroup Forum 68(3) (2004), 335–356.

[Do] C. Doss, Certain equivalence relations in transformation semi- groups, Ph.D. Thesis, Univ. of Tennessee, 1955.

[Fa] C. Faith, Ring theory. Grundlehren der Mathematischen Wissen- schaften, No. 191. Springer-Verlag, Berlin, 1976.

[Fe1] V. Fernandes, Semigroups of order preserving mappings on a finite chain: a new class of divisors. Semigroup Forum 54(2) (1997), 230–236.

[Fe2] V. Fernandes, A division theorem for the pseudovariety gener- ated by semigroups of orientation preserving transformations on a finite chain. Comm. Alg. 29(1) (2001), 451–456.

[Fe3] V. Fernandes, The monoid of all injective order preserving partial transformations on a finite chain. Semigroup Forum 62(2) (2001), 178–204.

[FP] D. FitzGerald, G. Preston, Divisibility of binary relations. Bull. Aust. Math. Soc. 5 (1971), 75–86.

[GK1] O. Ganyushkin, T. Kormysheva, The chain decomposition of partial permutations and classes of conjugate elements of the semigroup ISn. (Ukrainian) Visnyk of Kyiv University, 1993, no. 2, 10–18. BIBLIOGRAPHY 285

[GK2] O. Ganyushkin, T. Kormysheva, Isolated and nilpotent subsemi- groups of a finite inverse symmetric semigroup. Dopov./Dokl. Akad. Nauk Ukr. (9) (1993), 5–9.

[GK3] O. Ganyushkin, T. Kormysheva, On nilpotent subsemigroups of a finite symmetric inverse semigroup. Mat. Zametki 56(3) (1994), 29–35, 157; translation in Math. Notes 56(3–4) (1994), 1023–1029 (1995).

[GK4] O. Ganyushkin, T. Kormysheva, The structure of nilpotent sub- semigroups of a finite inverse symmetric semigroup. Dopov. Nats. Akad. Nauk Ukr. 1995, (1) 8–10.

[GM1] O. Ganyushkin, V. Mazorchuk, The structure of subsemigroups of factor powers of finite symmetric groups. Mat. Zametki 58(3) (1995), 341–354, 478; translation in Math. Notes 58(3–4) (1995), 910–920 (1996).

[GM2] O. Ganyushkin, V. Mazorchuk, On the structure of IOn. Semi- group Forum 66(3) (2003), 455–483.

[GM3] O. Ganyushkin, V. Mazorchuk, L-andR-cross-sections in ISn. Comm. Alg. 31(9) (2003), 4507–4523.

[GM4] O. Ganyushkin, V. Mazorchuk, Combinatorics of nilpotents in symmetric inverse semigroups. Ann. Combin. 8(2) (2004), 161–175.

[GM5] O. Ganyushkin, V. Mazorchuk, Combinatorics and distributions of partial injections. Australas. J. Combin. 34 (2006), 161–186.

[GM6] O. Ganyushkin, V. Mazorchuk, On classification of maximal nilpotent subsemigroups. J. Algebra 320 (2008), no. 8, 3081–3103.

[GMS] O. Ganyushkin, V. Mazorchuk, B. Steinberg, On the irreducible representations of a finite semigroup, Preprint arXiv:0712.2076.

[GP] O. Ganyushkin, M. Pavlov, On the cardinalities of a class of nilpo- tent semigroups and their automorphism groups, In: Algebraic structures and their applications, Proceedings of the Ukrainian Mathematical Congress 2001, Kyiv, Institute of Mathematics of the National Academy of Sciences of Ukraine, 2002, 17–21.

[GTS] O. Ganyushkin, O. Temnikov, G. Shafranova (Kudryavtseva), Groups of automorphisms for maximal nilpotent subsemigroups of the semigroup IS(M). (Ukrainian) Mat. Stud. 13(1) (2000), 11–22. 286 BIBLIOGRAPHY

[Ga1] G. Garba, Nilpotents in semigroups of partial one-to-one order- preserving mappings. Semigroup Forum 48(1) (1994), 37–49. [Ga2] G. Garba, Nilpotents in semigroups of partial order-preserving transformations. Proc. Edinburgh Math. Soc. (2) 37(3) (1994), 361–377. [GH1] G. Gomes, J. Howie, Nilpotents in finite symmetric inverse semi- groups. Proc. Edinburgh Math. Soc. (2) 30(3) (1987), 383–395. [GH2] G. Gomes, J. Howie, On the ranks of certain semigroups of order- preserving transformations. Semigroup Forum 45(3) (1992), 272–282. [Gr] J. Green, On the structure of semigroups. Ann. Math. 54 (1951), 163–172. [Gri] P. Grillet, Semigroups. An introduction to the structure theory. Monographs and Textbooks in Pure and Applied Mathematics, 193. Marcel Dekker, New York, 1995. [Grm] R. Grimaldi, Discrete and combinatorial mathematics, an applied introduction, 4th ed. Addison Wesley Longman, 2000. [Grd] C. Grood, A Specht module analog for the rook monoid. Electron. J. Combin. 9(1) (2002), Research Paper 2, 10 pp. [HS] B. Harris, L. Schoenfeld, The number of idempotent elements in symmetric semigroups. J. Combin. Theory 3 (1967), 122–135.

[HZ1] E. Hewitt, H. Zuckerman, The l1-algebra of a commutative semi- group. Trans. Amer. Math. Soc. 83 (1956), 70–97. [HZ2] E. Hewitt, H. Zuckerman, The irreducible representations of a semi-group related to the symmetric group. Illinois J. Math. 1 (1957), 188–213. [Hic1] J. Hickey, Semigroups under a sandwich operation. Proc. Edin- burgh Math. Soc. (2) 26(3) (1983), 371–382. [Hic2] J. Hickey, On variants of a semigroup. Bull. Aust. Math. Soc. 34(3) (1986), 447–459. [Hi1] P. Higgins, Techniques of semigroup theory. Oxford Science Pub- lications. The Clarendon Press, Oxford University Press, New York, 1992. [Hi2] P. Higgins, Combinatorial results for semigroups of order- preserving mappings. Math. Proc. Cambridge Philos. Soc. 113(2) (1993), 281–296. BIBLIOGRAPHY 287

[Hi3] P. Higgins, Idempotent depth in semigroups of order-preserving mappings. Proc. Roy. Soc. Edinburgh Sect. A 124(5) (1994), 1045–1058.

[HMR] P. Higgins, J. Mitchell, N. Ruˇskuc, Generating the full transforma- tion semigroup using order preserving mappings. Glasgow Math. J. 45(3) (2003), 557–566.

[Hoe] O. H¨older, Bildung zusammengesetzter Gruppen. Math. Ann. 46 (1895), 321–422.

[Ho1] J. Howie, The subsemigroup generated by the idempotents of a full transformation semigroup. J. London Math. Soc. 41 (1966), 707–716.

[Ho2] J. Howie, Products of idempotents in certain semigroups of transformations. Proc. Edinburgh Math. Soc. (2) 17 (1970/71), 223–236.

[Ho3] J. Howie, An introduction to semigroup theory. L.M.S. Mono- graphs, No. 7. Academic Press, London, 1976.

[Ho4] J. Howie, Semigroups and combinatorics. Monash Conference on Semigroup Theory (Melbourne, 1990), 135–139. World Sci. Publ., River Edge, NJ, 1991.

[Ho5] J. Howie, Combinatorial and arithmetical aspects of the theory of transformation semigroups. Seminario do Centro de Algebra, University of Lisbon (1992), 1–14.

[Ho6] J. Howie, Combinatorial and probabilistic results in transforma- tion semigroups. Words, languages and combinatorics, II (Kyoto, 1992), 200–206. World Sci. Publ., River Edge, NJ, 1994.

[Ho7] J. Howie, Fundamentals of semigroup theory. London Mathemati- cal Society Monographs. New Series, 12. Oxford University Press, London, 1995.

[HM] J. Howie, R. McFadden, Ideals are greater on the left. Semigroup Forum 40(2) (1990), 247–248.

[II] Na. Iwahori, No. Iwahori, On a set of generating relations of the full transformation semigroups. J. Combin. Theory Ser. A 16 (1974), 147–158.

[JM] S. Janson, V. Mazorchuk, Some remarks on the combinatorics of ISn. Semigroup Forum 70(3) (2005), 391–405. 288 BIBLIOGRAPHY

[Ji] C. Jianmiao, On sandwich semigroups in Boolean group algebras. J. Pure Appl. Alg. 169(2–3) (2002), 249–265. [KM] M. Kargapolov, J. Merzljakov, Fundamentals of the theory of groups. Graduate Texts in Mathematics, 62. Springer-Verlag, Berlin, 1979. [KS] M. Katsura, T. Saito, Maximal inverse subsemigroups of the full transformation semigroup. Semigroups with applications, 101– 113. World Sci. Publ., River Edge, NJ, 1992. [Ka] L. Katz, Probability of indecomposability of a random mapping function. Ann. Math. Stat. 26 (1955), 512–517. [KL] T. Khan, M. Lawson, Variants of regular semigroups. Semigroup Forum 62(3) (2001), 358–374. [KRS] D. Kleitman, B. Rothschild, J. Spencer, The number of semi- groups of order n. Proc. Amer. Math. Soc. 55(1) (1976), 227–232. [Ko] G. K¨othe, Uber¨ maximale nilpotente Unterringe und Nilringe. Math. Ann. 103(1) (1930), 359–363. [Ku1] G. Kudryavtseva, The structure of automorphism groups of semi- group inflations. Alg. Discrete Math. 6(1) (2007), 62–67. [Ku2] G. Kudryavtseva, On conjugacy in regular epigroups. Preprint arXiv:math/0605698. [KuMa1] G. Kudryavtseva, V. Mazorchuk, On conjugation in some trans- formation and Brauer-type semigroups. Publ. Math. Debrecen 70(1–2) (2007), 19–43. [KuMa2] G. Kudryavtseva, V. Mazorchuk, On presentations of Brauer-type monoids. Cent. Eur. J. Math. 4(3) (2006), 413–434. [KuMa3] G. Kudryavtseva, V. Mazorchuk, On the semigroup of square ma- trices, Alg. Colloq. 15(1) (2008), 33–52. [KuMa4] G. Kudryavtseva, V. Mazorchuk, On three approaches to conju- gacy in semigroups, Preprint arXiv:0709.4341, to appear in Semi- group Forum. [KuMa5] G. Kudryavtseva, V. Mazorchuk, Schur-Weyl dualities for symmetric inverse semigroups. J. Pure Appl. Algebra 212 (2008), no. 8, 1987–1995. [KuMa6] G. Kudryavtseva, V. Mazorchuk, Combinatorial Gelfand mod- els for some semigroups and q-rook monoid algebras, Preprint arXiv:0710.1972, to appear in Proc. Edinburgh Math. Soc. BIBLIOGRAPHY 289

[KT] G. Kudryavtseva, G. Tsyaputa, The automorphism group of the sandwich inverse symmetric semigroup. (Ukrainian) Bulletin of the University of Kiev, Series: Mechanics and Mathematics 2005, no. 13–14, 101–105.

[KMM] G. Kudryavtseva, V. Maltcev, V. Mazorchuk, L-andR-cross- sections in the Brauer semigroup. Semigroup Forum 72(2) (2006), 223–248.

[La] E. Landau, Uber¨ die Maximalordnung der Permutation gegebenen Grades. Arch. Math. Phys. Ser. 3. 5 (1903), 92–103.

[LU1] A. Laradji, A. Umar, On the number of nilpotents in the partial symmetric semigroup. Comm. Alg. 32(8) (2004), 3017–3023.

[LU2] A. Laradji, A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations. J. Integer Seq. 7(3) (2004), Article 04.3.8, 14 pp. (electronic).

[LU3] A. Laradji, A. Umar, Combinatorial results for semigroups of order-preserving partial transformations. J. Alg. 278(1) (2004), 342–359.

[LU4] A. Laradji, A. Umar, On certain finite semigroups of order- decreasing transformations. I. Semigroup Forum 69(2) (2004), 184–200.

[LU5] A. Laradji, A. Umar, Combinatorial results for semigroups of order-preserving full transformations. Semigroup Forum 72(1) (2006), 51–62.

[LU6] A. Laradji, A. Umar, Asymptotic results for semigroups of order- preserving partial transformations. Comm. Alg. 34(3) (2006), 1071–1075.

[Law] M. Lawson, Inverse semigroups. The theory of partial symmetries. World Sci. Publ., River Edge, NJ, 1998.

[Le] I. Levi, Congruences on normal transformation semigroups. Math. Japon. 52(2) (2000), 247–261.

[Lev] J. Levitzki, Uber¨ nilpotente Unterringe. Math. Ann. 105(1) (1931), 620–627.

[Lib] A. Liber, On symmetric generalized groups. Mat. Sb. N.S. 33(75), (1953), 531–544. 290 BIBLIOGRAPHY

[LPS] M. Liebeck, C. Praeger, J. Saxl, A classification of the maximal subgroups of the finite alternating and symmetric groups. J. Alg. 111(2) (1987), 365–383.

[Li] S. Lipscomb, Symmetric inverse semigroups. Mathematical Sur- veys and Monographs, 46. American Mathematical Society, Prov- idence, RI, 1996.

[LY] C. Lu, X. Yang, Maximal properties of some subsemigroups in finite order-preserving transformation semigroups. Comm. Alg. 28(7) (2000), 3125–3135.

[Ly] E. Lyapin, Semigroups. (Russian) Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1960.

[MS1] K. Magill, Jr., S. Subbiah, Green’s relations for regular elements of sandwich semigroups. I. General results. Proc. London Math. Soc. (3) 31(2) (1975), 194–210.

[MS2] K. Magill, Jr., S. Subbiah, Green’s relations for regular elements of sandwich semigroups. II. Semigroups of continuous functions. J. Aust. Math. Soc. Ser. A 25(1) (1978), 45–65.

[MMT] K. Magill, Jr., P. Misra, U. Tewari, Structure spaces for sandwich semigroups. Pacific J. Math. 99(2) (1982), 399–412.

[MR] M. Malandro, D. Rockmore, Fast Fourier transforms for the rook monoid, Preprint arXiv:0709.4175.

[Ma1] A. Mal’cev, Symmetric groupoids. Mat. Sb. N.S. 31(73), (1952), 136–151.

[Ma2] A. Mal’cev, Nilpotent semigroups. Ivanov. Gos. Ped. Inst. Uchen. Zap. Fiz.-Mat. Nauki 4 (1953), 107–111.

[MT] V. Mazorchuk, G. Tsyaputa, Isolated subsemigroups in the vari- ants of Tn. Acta Math. Univ. Com., Vol. LXXVII, 1 (2008), 63–84.

[Mi] W. Miller, The maximum order of an element of a finite symmetric group. Amer. Math. Monthly 94(6) (1987), 497–506.

[Mo1] M. Mogilevski˘ı, Order relations on symmetric semigroups of trans- formations and on their homomorphic images. Semigroup Forum 19(4) (1980), 283–305.

[Mo2] M. Mogilevski˘ı, A remark on the orderings of the semigroup of total transformations. Ordered sets and lattices, No. 2, p. 59. Izdat. Saratov Univ., Saratov, 1974. BIBLIOGRAPHY 291

1 [Moo] E. Moore, Concerning the abstract groups of order k!and 2 k! holohedrically isomorphic with the symmetric and alternating substitution groups on k letters, Proc. London Math. Soc. 28 (1897), 357–366.

[Mu1] W. Munn, On semigroup algebras. Proc. Cambridge Philos. Soc. 51 (1955), 1–15.

[Mu2] W. Munn, Matrix representations of semigroups. Proc. Cambridge Philos. Soc. 53 (1957), 5–12.

[Mu3] W. Munn, The characters of the symmetric inverse semigroup. Proc. Cambridge Philos. Soc. 53 (1957), 13–18.

[Mu4] W. Munn, Irreducible matrix representations of semigroups. Quart. J. Math. Oxford Ser. (2) 11 (1960), 295–309.

[Ne] P. Neumann, A lemma that is not Burnside’s. Math. Sci. 4(2) (1979), 133–141.

[Ni] J. Nichols, A class of maximal inverse subsemigroups of TX . Semi- group Forum 13(2) (1976/77), 187–188.

[Pek1] V. Pekhterev, Retracts of the semigroup Tn. (Ukrainian) Visnyk of Kyiv University, 2003, no. 10, 127–129.

[Pek2] V. Pekhterev, H-andR-cross-sections of the full finite semigroup Tn. Alg. Discrete Math. 2(3) (2003), 82–88.

[Pek3] V. Pekhterev, R-cross-sections of the semigroup TX . (Ukrainian) Mat. Stud. 21(2) (2004), 133–139.

[Pek4] V. Pekhterev, H-, R-andL-cross-sections of the infinite sym- metric inverse semigroup ISX . Alg. Discrete Math. 4(1) (2005), 92–104.

[Pe] M. Petrich, Introduction to semigroups. Merrill Research and Lec- ture Series. Charles E. Merrill Publishing Co., Columbus, OH, 1973.

[Pi] J. Pin, Vari´et´es de langages formels. (French) With a preface by M. P. Sch¨utzenberger. Etudes´ et Recherches en Informatique. Masson, Paris, 1984. 160 pp.

[Po1] I. Ponizovskiy, Transitive representations by transformations of semi-groups of a certain class. (Russian) Sibirsk. Mat. Zh. 5 (1964), 896–903. 292 BIBLIOGRAPHY

[Po2] I. Ponizovskiy, Representations of inverse semigroups by partial one-to-one transformations. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 989–1002.

[Po3] I. Ponizovskiy, On matrix representations of associative systems. (Russian) Mat. Sb. N.S. 38 (80) (1956), 241–260.

[Po4] I. Ponizovskiy, Some examples of semigroup algebras of finite rep- resentation type. (Russian) Zap. Nauchn. Sem. Leningrad. Ot- del. Mat. Inst. Steklov. (LOMI) 160 (1987), Anal. Teor. Chisel i Teor. Funktsii. 8, 229–238, 302; translation in J. Sov. Math. 52(3) ¯ (1990), 3170–3178.

[Pp] L. Popova, Defining relations of certain semigroups of partial transformations of a finite set. (Russian) Leningrad. Gos. Ped. Inst. Uchen. Zap. 218 (1961), 191–212.

[Pu1] M. Putcha, Complex representations of finite monoids. Proc. London Math. Soc. (3) 73(3) (1996), 623–641.

[Pu2] M. Putcha, Complex representations of finite monoids. II. Highest weight categories and quivers. J. Alg. 205(1) (1998), 53–76.

[Pu3] M. Putcha, Reciprocity in character theory of finite semigroups. J. Pure Appl. Alg. 163(3) (2001), 339–351.

[Re1] N. Reilly, Embedding inverse semigroups in bisimple inverse semi- groups. Quart. J. Math. Oxford Ser. (2) 16 (1965), 183–187.

(1) [Re2] N. Reilly, Maximal inverse subsemigroups of TX . Semigroup Fo- rum 15(4) (1977/78), 319–326.

[Ri] C. Ringel, The representation type of the full transformation semi- group T4. Semigroup Forum 61(3) (2000), 429–434.

[Sa] B. Sagan, The symmetric group. Representations, combinatorial algorithms, and symmetric functions. Second edition. Graduate Texts in Mathematics, 203. Springer-Verlag, New York, 2001.

[SYT] S. Satoh, K. Yama, M. Tokizawa, Semigroups of order 8. Semi- group Forum 49(1) (1994), 7–29.

[Sc1] B. Schein, Representations of generalized groups. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 28(3) (1962), 164–176.

[Sc2] B. Schein, Representation of semigroups by binary relations. (Russian) Dokl. Akad. Nauk SSSR 142 (1962), 808–811. BIBLIOGRAPHY 293

[Sc3] B. Schein, Transitive representations of semigroups. (Russian) Usp. Mat. Nauk 18(3) (1963), 215–222.

[Sc4] B. Schein, Lectures on transformation semigroups. (Russian) Spe- cial course. Izdat. Saratov Univ., Saratov, 1970.

[Sc5] B. Schein, A symmetric semigroup of transformations is covered by its inverse subsemigroups. Acta Math. Acad. Sci. Hung. 22 (1971/72), 163–171.

[ST1] B. Schein, B. Teclezghi, Endomorphisms of finite symmetric in- verse semigroups. J. Alg. 198(1) (1997), 300–310.

[ST2] B. Schein, B. Teclezghi, Endomorphisms of finite full transforma- tion semigroups. Proc. Amer. Math. Soc. 126(9) (1998), 2579– 2587.

[ST3] B. Schein, B. Teclezghi, Endomorphisms of symmetric semigroups of functions on a finite set. Comm. Alg. 26(12) (1998), 3921–3938.

[Sch] J. Schreier, Uber¨ Abbildungen einer abstrakten Menge auf ihre Teilmengen. Fundam. Math. 28 (1936), 261–264.

[Sw1] S.ˇ Schwarz, The theory of characters of finite commutative semi- groups. (Russian) Czech. Math. J. 4(79) (1954), 219–247.

[Sw2] S.ˇ Schwarz, Characters of commutative semigroups as class func- tions. (Russian) Czech. Math. J. 4(79) (1954), 291–295.

[Sw3] S.ˇ Schwarz, On a Galois connexion in the theory of characters of commutative semigroups. (Russian) Czech. Math. J. 4(79) (1954), 296–313.

[Sh1] G. Shafranova, Maximal nilpotent subsemigroups of the semi- group PAutq(Vn). Probl. Alg. 13 (1998), 69–83. [Sh2] G. Shafranova, Nilpotent subsemigroups of transformation semi- groups. (Ukrainian) Ph.D. Thesis, Kyiv University, Kyiv, Ukraine, 2000.

[She] L. Shevrin, On nilsemigroups. (Russian) Usp. Mat. Nauk 14(5) (1959), 216–217.

[Sol] A. Solomon, Monoids of order-preserving transformations of a fi- nite chain, Res. report 94-8, School of Math. and Stat., University of Sydney, 1994.

[So] L. Solomon, Representations of the rook monoid. J. Alg. 256(2) (2002), 309–342. 294 BIBLIOGRAPHY

[Sta1] R. Stanley, Enumerative combinatorics. Vol. 2. Cambridge Stud- ies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999.

[Sta2] R. Stanley, Catalan addendum, manuscript, available at: http://www-math.mit.edu/˜rstan/ec/catadd.pdf

[Ste1] B. Steinberg, M¨obius functions and semigroup representation the- ory. J. Combin. Theory Ser. A 113(5) (2006), 866–881.

[Ste2] B. Steinberg, M¨obius functions and semigroup representa- tion theory II: Character formulas and multiplicities, Preprint arXiv:math/0607564, to appear in Adv. Math.

[St1] G. Stronska, Nilpotent subsemigroups of the semigroup of order- decreasing transformations, Mat. Stud. 22(2) (2004), 184–197.

[St2] G. Stronska, On the automorphisms for the nilpotent subsemi- group of the order-decreasing transformation semigroup. Proceed- ings of Gomel University, 2007.

[Su1] A. Suschkewitsch, Untersuchunguber ¨ verallgemeinerte Substitu- tionen. Atti del Congresso Internazionale dei Matematici Bologna, 1 (1928), 147–157.

[Su2] A. Suschkewitsch, Theory of generalized groups. (Russian) Kharkov-Kyiv, GNTI, 1937

[Sut1] E. Sutov, Defining relations in finite semigroups of partial trans- formations. Dokl. Akad. Nauk SSSR 132 1280–1282 (Russian); translated as Sov. Math. Dokl. 1 (1960), 784–786.

[Sut2] E. Sutov, Homomorphisms of the semigroup of all partial transfor- mations. Izv. Vyssh. Uchebn. Zaved. Mat. 22(3) (1961), 177–184.

[Sy1] J. Symons, On a generalization of the transformation semigroup. J. Aust. Math. Soc. 19 (1975), 47–61.

[Sy2] J. Symons, Some results concerning a transformation semigroup. J. Aust. Math. Soc. 19(4) (1975), 413–425.

∗ [Sz] M. Szalay, On the maximal order in Sn and Sn. Acta Arith. 37 (1980), 321–331.

[Ts1] G. Tsyaputa, Transformation semigroups with the deformed mul- tiplication. (Ukrainian) Bulletin of the University of Kiev, Series: Physics and Mathematics, 2003, no. 4, 82–88. BIBLIOGRAPHY 295

[Ts2] G. Tsyaputa, Green’s relations on the deformed transformation semigroups. Alg. Discrete Math. 3(1) (2004), 121–131.

[Ts3] G. Tsyaputa, Deformed multiplication in the semigroup PT n. (Ukrainian) Bulletin of the University of Kiev, Series: Mechan- ics and Mathematics, 2004, no. 11–12, 35–38. [Ts4] G. Tsyaputa, Isolated and nilpotent subsemigroups in the variants of ISn. Alg. Discrete Math. 5(1) (2006), 89–97. [Um1] A. Umar, Semigroups of order-decreasing transformations, Ph.D. Thesis, University of St. Andrews, 1992. [Um2] A. Umar, On the semigroups of order-decreasing finite full trans- formations. Proc. Roy. Soc. Edinburgh Sect. A 120(1–2) (1992), 129–142. [Um3] A. Umar, On the semigroups of partial one-to-one order- decreasing finite transformations. Proc. Roy. Soc. Edinburgh Sect. A 123(2) (1993), 355–363. [Vl1] T. Voloshyna, Effective transitive representations of the finite symmetric inverse semigroup. (Ukrainian) Visn. Kyiv Univ. 1998(2) (1998), 16–21. [Vl2] T. Voloshyna, Closed inverse subsemigroups of monogenic inverse semigroups. (Ukrainian) Visn. Kyiv Univ. 2000(3), 16–23. [Vl3] T. Voloshyna, Effective transitive representations of monogenic inverse semigroups. (Ukrainian) Mat. Stud. 16(1) (2001), 25–36. [Vo1] N. Vorob’ev, On ideals of associative systems. Dokl. Akad. Nauk SSSR (N.S.) 83 (1952), 641–644. [Vo2] N. Vorob’ev, On symmetric associative systems. Leningrad. Gos. Ped. Inst. Uchen. Zap. 89 (1953), 161–166. [H.Ya] H. Yang, Two notes of maximal inverse subsemigroups of full trans- formation semigroups. J. Math. (Wuhan) 24(4) (2004), 375–380. [X.Ya1] X. Yang, A classification of maximal inverse subsemigroups of the finite symmetric inverse semigroups. Comm. Alg. 27(8) (1999), 4089–4096. [X.Ya2] X. Yang, A classification of maximal subsemigroups of finite order-preserving transformation semigroups. Comm. Alg. 28(3) (2000), 1503–1513. [YY] H.Yang, X. Yang, L (or R)-cross-sections of finite symmetric in- verse semigroups, Preprint 2002. List of Notation

(Γ˜1, Γ˜2,...,Γ˜k) Linear notation for a connected element with a cycle

(x0,x1,...,xk−1) Oriented cycle

[Γ˜1, Γ˜2,...,Γ˜k; a] Linear notation for a tree with a sink

[x]m The polynomial x(x − 1)(x − 2) ...(x − m +1) α A transformation

α(x)Thevalueofα at x

α(x)=∅ α is not defined at x

α : M → Mαis a transformation of M

α|B The restriction of α to B α(K) Restriction of α to the orbit K i i i i ··· αj An element of the form [u1,u2,...,uj](a)(b) (c)

αk The element [1, 2,...,k](k +1)···(n) i i i i ··· βj An element of the form [uj,uj+1,...,umi ](a)(b) (c)

(i) ηi The canonical identification of M and M

Γα The graph of α

ιS The identity congruence on S λ  nλis a partition of n

λu The mapping x → xu A The set of all elements generated by A

a A cyclic semigroup, generated by a

A|Σ Semigroup generated by A with defining relations Σ

297 298 LIST OF NOTATION

C[S] The semigroup algebra of S over C

Ctriv The trivial S-module N The set of all positive integers

Zy The set of all residue classes modulo y 0 The nowhere defined partial transformation

0n The nowhere defined partial transformation of N

GLn The group of invertible n × n matrices

KN The full directed graph on N m A partition of N into a disjoint union of nonempty subsets

N The set {1, 2,...,n}

Nn The set {1, 2,...,n} Z(S) The center of the semigroup S

Ak The alternating group B(X) The Boolean of X C S Sn (α) The centralizer of α in n D Green’s relation D

D(a)TheD-class of a

Dk The set of all elements of rank k in Tn, PT n,orISn E(S) The set of idempotents of S

Ek The trivial subgroup of Sk

Fn The set of all order-decreasing total transformations H Green’s relation H

H(a)TheH-class of a

IFn The set of all order-decreasing partial permutations

IOn The set of all order-preserving partial permutations

IOn The set of all order-preserving partial permutations IO≺ ≺ n The set of all -order-preserving partial permutations LIST OF NOTATION 299

ISn The symmetric inverse semigroup on N

Ik The set of all elements of rank at most k I∗ k The dual symmetric inverse semigroup

Iρ The unique ideal of the congruence ρ J Green’s relation J

J (a)TheJ -class of a

L Green’s relation L

L(a)TheL-class of a

M(e, H, a) The set used to construct transitive actions by partial transfor- mations

Nn The set of all nilpotent elements of ISn

On The set of all order-preserving total transformations

PFn The set of all order-decreasing partial transformations

POn The set of all order-preserving partial transformations PT (M) The set of all partial transformations of M

PT n The set of all partial transformations of N R Green’s relation R

R(a)TheR-class of a

SPOn Semigroup of strictly partial order-preserving transformations of N

Sn The symmetric group on N T (M) The set of all total transformations of M

Tn The set of all total transformations of N

V4 The Klein 4-group as a subgroup of S4 B The bicyclic semigroup h Congruence associated with variants in The canonical inclusion of PT n to Tn+1

On The set of all partial orders on N 300 LIST OF NOTATION oα(x) The orbit of x in Γα p(n) The partition function rn The number of Schr¨oder paths of order n t(α)Thetypeofα tk(α)Thenumberofx ∈ N for which |{y ∈ N : α(y)=x}| = k

0a The constant transformation with the image {a}

Annl(A) The left annihilator of the set A

Annl(a) The left annihilator of the element a

Annr(a) The right annihilator of the element a

Annr(A) The right annihilator of the set A Aut(S) The set of all automorphisms of S ct(π) The cyclic type of the permutation π def(α) The defect of α dom(α) The domain of α

End(S) The set of all endomorphisms of S

fi(g) The number of fixed points of g

HomS(V,W) ThesetofallS-homomorphisms from V to W idrank(S) Idempotent rank of the semigroup S im(α) The image of α

Inn(S) The set of all inner automorphisms of S invσ The number of inversions for σ Ker(ϕ) The kernel of the homomorphism ϕ lcm The least common multiple

Matn(C) The algebra of all complex n × n matrices min(A, ≺) The minimal element of A with respect to ≺ nd(a) The nilpotency degree of a nd(S) The nilpotency degree of a nilpotent semigroup S LIST OF NOTATION 301 nilrank(S) Nilpotent rank of the semigroup S

Nil(S) The set of all nilpotent subsemigroups of S

Nilk(S) The set of all nilpotent subsemigroups of S of nilpotency degree k rank(α) The rank of α rank(S) Rank of the semigroup S stim(α) The stable image of α strank(α) The stable rank of α

StG(m) stabilizer of the point m with respect to the action of G trivS trivial action of a semigroup S trα(x) The trajectory of x in Γα

Bn The nth Bell number

Cn The n-th Catalan number

Cn The n-th Catalan number

In The cardinality of ISn

Ln The total number of chains in the chain decompositions of all ele- ments of ISn

Mn The total number of chains in the chain decompositions of all nilpo- tent elements of ISn n(PT n) The total number of nilpotent elements in PT n

Nn The total number of nilpotent elements in ISn

Nn The total number of nilpotent elements in ISn

OG The number of G-orbits

Pn The total number of fixed points of all elements of ISn S(n, k) Stirling numbers of the second kind s(n, k) Stirling numbers of the first kind

ωS The uniform congruence on S

ωα Binary relation describing orbits of α + The addition of residue classes 302 LIST OF NOTATION dom(α) The codomain of α a The equivalence class of a aρ The equivalence class of a with respect to the equivalence relation ρ

Kn The semigroup, generated by α1,...,αn in Section 12.5 x The residue class of x ← ≺ The linear order opposite to ≺ ← S The dual of S ∪{ } πα A special binary relation on N n +1

πk The transposition (1,k)

πα A special binary relation on N

πρ The canonical projection S  S/ρ  Partial order on the set of all partitions of N

ρ(S, A) Kernel of the natural epimorphism A+  S

ρI The Rees congruence with respect to the ideal I

ρα A partition of N, associated to α

ρΣ The minimal congruence containing all relations from Σ

B ΣA System of relations for B induced from that for A

∼pS Relation of primary S-conjugation

∼S Relation of S-conjugation √ e The set of all x such that xm = e for some m

τT A binary relation on N associated with a nilpotent subsemigroup T of PT n

Υi exceptional endomorphisms Δ The equality relation

ε The identity transformation

εA The unique idempotent of ISn with domain A

εn The identity transformation on N LIST OF NOTATION 303

ε(k) The idempotent ε{1,...,k−1,k,...,n} · { } εi,J The product εi,j1 εi,j2 εi,jk , where J = j1,...,jk

εm,k The idempotent of rank (n − 1) satisfying εm,k(m)=εm,k(k)=m ∗ Λa The inner automorphisms corresponding to a ∈ S

Ωϕ An exceptional endomorphism

Φα The binary relation associated with α

Φ An endomorphism of rank one

Ψ ,δ An endomorphism of rank two

τ Θ ,δ An endomorphism of rank three |a| The order of the element a

|S| The cardinality of S

Ξ An exceptional endomorphism

ξ ◦ η Product of binary relations ξ and η aρb The pair (a, b) belongs to the binary relation ρ a · b The product of a and b a ≡ bais equivalent to b a ≡ b(ρ) a is equivalent to b with respect to the equivalence relation ρ aDbaand b are D-equivalent aHbaand b are H-equivalent aJ baand b are J -equivalent aLbaand b are L-equivalent aRbaand b are R-equivalent a ∼pS baand b are primarily S-conjugate a ∼S baand b are S-conjugate A+ The set of all (finite) words over the alphabet A a−1 The inverse of a in an inverse semigroup a−1 The inverse of a 304 LIST OF NOTATION

AB The product of the sets A and B ab The product of a and b

Eα Arrows of Γα F (X, Y ) The set of all mappings from X to Y f : X → Y A mapping from X to Y

G/H The set of all left cosets of G modulo H g · m The element ϕ(g)(m) for the action ϕ of G on M

Ge The maximal subgroup corresponding to the idempotent e HG His a normal subgroup of G

Kn The semigroup Kn ∪{ε}

L(≺1,...,≺k)AnL-cross-section of ISn n A nonnegative integer IS Nτ The nilpotent subsemigroup of n associated to the partial order τ on N

Nτ The nilpotent subsemigroup of PT n associated to the partial order τ on N

R(≺1,...,≺k)AnR-cross-section of ISn S/ρ The quotient of the semigroup S modulo the congruence ρ

S/I The quotient of the semigroup S modulo the Rees congruence ρI ∼ S = TSis isomorphic to T S∗ The group of units of the semigroup S

S0 S if 0 ∈ S,andS ∪{0} otherwise

S1 S if 1 ∈ S,andS ∪{1} otherwise

Sa The variant of S with respect to the sandwich element a

k S {a1a2 ···ak : ai ∈ S, i =1,...,k}, k>1 T

V (M) S-module L-induced from M ∼ V = WVis isomorphic to W LIST OF NOTATION 305

VS(a) The set of elements, inverse to a ∈ S

Vα Vertices of Γα x mod y The residue of x modulo y Index

G-conjugate elements, 103 annihilator G-orbit, 151 left, 65 S-conjugate elements, 104 right, 65 primarily, 104 anti-chain, 51 S-homomorphism, 191 arrow, 3 S-module, 189 head of, 3 Σ-pair, 154 tail of, 3 λ-tableau, 206 automorphism, 93, 111 standard, 206 extendable, 248 H-cross-section, 219 inner, 111 ρ-cross-section, 215 band action rectangular, 36 centralizer of, 210 Bell number, 48, 138 action of group, 175 bicyclic semigroup, 246 kernel of, 177 binary operation, 16 stabilizer, 177 associative, 16 action of semigroup, 175 Boolean, 35 by partial permutations, 175 braid relations, 172 by partial transformations, 175 Branching rule, 207 by transformations, 175 equivalent, 175 canonical epimorphism, 93 faithful, 176 canonical form, 154 invariant subset, 176 canonical projection, 93 left, 186 Catalan number, 152, 254 natural, 176 Cayley table, 20 quasi-transitive, 176 Cayley’s Theorem, 21 right, 186 center, 129 semitransitive, 187 centralizer, 121, 210 similar, 175 chain, 23 transitive, 176 prefix of, 229 trivial, 176 chain-cycle notation, 23 alphabet, 153 closure word over, 153 transitive, 104 alternating group, 108 concatenation, 153 307 308 INDEX congruence, 91 H-equivalent, 55 identity, 91 J -equivalent, 55 left, 91 L-equivalent, 53 Rees, 92 R-equivalent, 54 right, 91 conjugate, 103 uniform, 91 generated by a set, 39 conjugate elements, 103 group, 70 connected component, 4 index of, 71 coset inverse, 23 left, 95 pair, 23 Coxeter group, 172 inverse of, 18 cross-section, 215 invertible, 18 cycle, 4, 6 mididentity, 249 oriented, 4 nil-, 28 cyclic semigroup, 70 nilpotency class, 28 diagram nilpotency degree, 28 egg-box, 56 nilpotent, 28 digraph, 3 order of, 71 direct sum of modules, 200 period of, 71 trivial, 200 regular, 23 directed edge, 3 sandwich, 237 directed graph, 3 singular, 75 arrow of, 3 type of, 71 head of, 3 unit, 18 tail of, 3 endomorphism, 93, 114 connected, 4 rank of, 114 directed edge of, 3 epimorphism, 93 isomorphic, 13 canonical, 93 oriented path of, 4 equivalent subgraph of, 4 D-, 54 union of, 7 H-, 55 vertex of, 3 J -, 55 dual, 33 L-, 53 duality R-, 54 Schur-Weyl, 210 effective representation, 198 Ferrers diagram, 206 egg-box diagram, 56 shape of, 206 element Fibonacci number, 258 G-conjugate, 103 forest, 6 S-conjugate, 104 free semigroup, 153 primarily, 104 base of, 153 D-equivalent, 54 full transformation semigroup, 16 INDEX 309 function constant on a set, 76 partition, 242 singular on a set, 76 identity, 17 Gelfand model, 210 left, 17 generating set, 39 mid-, 249 generating system, 39 right, 17 Coxeter, 42 two-sided, 17 depth of, 269 image irreducible, 39 homomorphic, 94 generator stable, 33 Coxeter, 172 inclusion global semigroup, 83 canonical, 149 graph induced module, 194 empty, 70 inflation, 239 isomorphic, 103 interval, 253 strongly connected, 270 invariant, 26, 31 Green’s lemma, 56 inverse element, 23 Green’s relations, 55 inverse semigroup, 25 group, 18 involution, 210 action of, 175 irreducible representation, 189 kernel of, 177 isomorphic graphs, 103 stabilizer, 177 isomorphism, 20, 93 alternating, 108 element of juxtaposition, 153 conjugate, 103 sub- kernel, 32, 94 normal, 95 lemma symmetric, 19 Burnside’s, 151 group element, 70 Cauchy-Frobenius, 151 homomorphic image, 94 Green, 56 homomorphism, 92, 191 letter, 153 kernel of, 94 linear notation, 9 ideal, 45 mapping, 15 generator of, 45 composition of, 15 left, 45 partial, 15 one-sided, 45 composition of, 15 prime, 66 defined on, 15 principal, 45 product of, 15 reflexive, 66 product of, 15 right, 45 value of, 15 semiprime, 66 maximal nilpotent subsemigroup, two-sided, 45 132 idempotent, 26 mididentity, 249 310 INDEX module linear, 9 equivalent, 191 number induced, 194 Bell, 48, 138 isomorphic, 191 Catalan, 152, 254 regular, 192 Fibonacci, 258 Specht, 206 Lah tensor product of, 209 signless, 30 module over a semigroup, 189 Stirling L-induced, 194 first kind, 275 direct sum, 200 second kind, 29 equivalent, 191 homomorphism, 191 operation indecomposable, 200 sandwich, 237 induced, 194 orbit isomorphic, 191 kernel, 32 quotient, 190 order regular, 192 partial semisimple, 201 natural, 109 simple, 189 oriented path, 4 sub-, 189 trivial, 190 pair of inverse elements, 23 modules partial order indecomposable, 200 height of, 134 semisimple, 201 natural, 109 monoid, 17 partial transformation rook, 190 type of, 38 monomorphism, 93 partition, 206 morphism partition function, 242 auto-, 93 path endo-, 93 break of, 4 epi-, 93 head of, 4 homo-, 92 oriented, 4 iso-, 93 Schr¨oder, 268 mono-, 93 tail of, 4 permutation, 3 natural module, 190 ≺-order-preserving, 220 nil-semigroup, 148 cyclic type of, 105 nilpotency class, 131 even, 205 nilpotency degree, 131 inversion for, 205 nilpotent semigroup, 131 odd, 205 normal subgroup, 95 order-preserving, 220 notation partial, 22 chain-cycle, 23 permutational part, 33 cyclic, 8 power semigroup, 83 INDEX 311 prefix, 229 semigroup, 16 presentation, 154 D-class, 54 irreducible, 173 regular, 63 presentation of semigroup, 154 H-class, 55 irreducible, 173 H-cross-section of, 219 Preston-Wagner Theorem, 33 J -class, 55 projection L-class, 53 canonical, 93 R-class, 54 ρ-retract, 216 quotient module, 190 abelian, 28 rank action of, 175 stable, 33 equivalent, 175 rank of semigroup, 269 faithful, 176 idempotent, 269 invariant subset, 176 nilpotent, 269 left, 186 rectangular band, 36 natural, 176 regular D-class, 63 quasi-transitive, 176 regular element, 23 right, 186 regular module, 192 semitransitive, 187 relation similar, 175 binary transitive, 176 product of, 54 trivial, 176 relations algebra, 207 braid, 172 aperiodic, 253 representation, 189 automorphism of, 93, 111 completely reducible, 201 inner, 111 representation of a semigroup bicyclic, 246 completely reducible, 201 cardinality, 16 representation of semigroup, 189 Cayley table, 20 effective, 198 center of, 129 irreducible, 189 combinatorial, 253 residue, 71 commutative, 28 residue classes, 71 congruence on, 91 group of, 71 identity, 91 restriction, 26 left, 91 retraction, 129 quotient modulo, 92 rook monoid, 190 Rees, 92 right, 91 sandwich uniform, 91 element, 237 cross-section operation, 237 H-, 219 semigroup, 237 cross-section of, 215 Schr¨oder path, 268 cyclic, 70 Schur-Weyl duality, 210 defining relation, 154 312 INDEX

dual, 33 involution of, 210 element isomorphism of, 20 canonical form, 154 left zero, 35 element of module over, 189 G-conjugate, 103 L-induced, 194 S-conjugate, 104 direct sum, 200 D-equivalent, 54 equivalent, 191 H-equivalent, 55 homomorphism of, 191 J -equivalent, 55 indecomposable, 200 L-equivalent, 53 induced, 194 R-equivalent, 54 isomorphic, 191 group, 70 quotient, 190 index of, 71 regular, 192 order of, 71 semisimple, 201 period of, 71 simple, 189 regular, 23 sub-, 189 singular, 75 trivial, 190 type of, 71 monomorphism of, 93 endomorphism of, 93, 114 morphism of rank of, 114 auto-, 93, 111 retraction, 129 endo-, 93, 114 epimorphism of, 93 epi-, 93 canonical, 93 iso-, 20 factor, 92 mono-, 93 free, 153 multiplication table, 20 base of, 153 nil-, 148 full transformation, 16 nilpotency class, 131 global, 83 nilpotency degree, 131 Green’s relations, 55 nilpotent, 131 homomorphism of, 92 class of, 131 kernel of, 94 degree of, 131 ideal of, 45 null, 35 generator of, 45 of all partial transformations, 16 left, 45 of all transformations, 16 one-sided, 45 of left identities, 35 prime, 66 of right identities, 35 principal, 45 opposite, 33 reflexive, 66 order-decreasing, 252 right, 45 order-preserving, 251 semiprime, 66 power, 83 two-sided, 45 presentation, 154 idempotent of, 26 irreducible, 173 inflation, 239 quotient, 92 inverse, 25 Rees, 92 INDEX 313

rank of, 269 signless Lah number, 30 idempotent, 269 simple module, 189 nilpotent, 269 singular part, 42 Rees quotient, 92 Specht module, 206 regular, 24 stable image, 33 relation, 153 stable rank, 33 contained in congruence, 154 subgraph, 4 defining, 154 maximal connected, 4 follows, 154 subgroup trivial, 155 coset modulo relation on left, 95 compatible, 54, 91 maximal, 69 Green’s, 55 normal, 95 left compatible, 54, 91 subsemigroup, 17 right compatible, 54, 91 Sn-normal, 108 representation of, 189 closed, 186 completely reducible, 201 completely isolated, 74 effective, 198 isolated, 74 irreducible, 189 completely, 74 retract of, 216 maximal, 84 right zero, 35 maximal nilpotent, 132 sandwich, 237 nilpotent self-dual, 33 maximal, 132 semisimple, 272 symmetric group, 19 singular part of, 42 sign module, 205 sub-, 17 symmetric inverse semigroup, 22 Sn-normal, 108 system closed, 186 generating, 39 completely isolated, 74 depth of, 269 isolated, 74 irreducible, 39 maximal, 84 subgroup of tensor product, 209 maximal, 69 theorem submodule, 189 Cayley, 21 symmetric, 16 orbit-counting, 151 symmetric inverse, 22 Preston-Wagner, 33 dual, 210 trajectory, 4 variant of, 237 terminates, 5 with zero multiplication, 35 transformation, 1 semigroup algebra, 207 Sn-centralizer of, 121 semisimple semigroup, 272 ≺-order-preserving, 220 set bijective, 3 generating, 39 chain, 23 sign module, 205 prefix of, 229 314 INDEX

codomain of, 2 stable rank, 33 constant, 20 surjective, 3 cycle of length k,10 tabular form of, 1 defect of, 2 total, 2 domain of, 1 type of, 38 fixed point of, 10 value of, 1 full, 2 transitive closure, 104 graph of, 3 transposition, 10 identity, 18 tree, 6 image forest of, 6 stable, 33 sink of, 6 image of, 2 trivial, 7 injective, 3 trivial block, 260 invariant set w.r.t., 26 trivial module, 190 orbit of, 6 order-decreasing, 252 unit, 18 order-increasing, 267 variant of semigroup, 237 order-preserving, 220, 251 order-reversing, 267 word, 153 order-turning, 268 concatenation of, 153 partial, 1 juxtaposition, 153 permutational part of, 33 preserves relation, 56 Young diagram, 206 range of, 2 shape of, 206 rank stable, 33 zero, 19 rank of, 2 left, 19 restriction of, 26 right, 19 stable image, 33 two-sided, 19