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The Class Equation and Counting in Factorizable Monoids

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The Class Equation and Counting in Factorizable Monoids PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 11, Pages 3345{3351 S 0002-9939(03)06920-X Article electronically published on February 14, 2003 THE CLASS EQUATION AND COUNTING IN FACTORIZABLE MONOIDS S. LIPSCOMB AND J. KONIECZNY (Communicated by Stephen D. Smith) Abstract. For orders and conjugacy in finite group theory, Lagrange's Theo- rem and the class equation have universal application. Here, the class equation (extended to monoids via standard group action by conjugation) is applied to factorizable submonoids of the symmetric inverse monoid. In particular, if M is a monoid induced by a subgroup G of the symmetric group Sn, then the center ZG(M)(allelementsofM that commute with every element of G)is Z(G)[f0g if and only if G is transitive. In the case where G is both transitive and of order either p or p2 (for p prime), formulas are provided for the order of M as well as the number and sizes of its conjugacy classes. 1. Introduction In this paper, we study monoids of restrictions of permutations under function composition. Indeed, for each subgroup G of the symmetric group Sn of all permu- tations of N = f1; 2;::: ;ng, we define the induced monoid M(G)=fα : α is a restriction of some σ 2 Gg whose multiplication is function composition. These monoids are inverse (for each α 2 M(G), there is a unique β 2 M(G) satisfying α = αβα and β = βαβ)and factorizable (each α 2 M(G)factorsα = εσ where " is an idempotent and σ 2 G is a unit). In the special case G = Sn, the induced monoid M(Sn) is denoted Cn, and it is widely known as the symmetric inverse monoid. Abstract factorizable inverse monoids have been studied by Chen and Hsieh [1]. The definition of the property of being factorizable (given originally for inverse monoids) has been generalized to arbitrary semigroups: A semigroup S is called factorizable if S = GE for some subgroup G of S and some set E of idempotents of S [7]. For example, the semigroup PTX of all partial transformations on a set X is factorizable if and only if X is finite [7]. A similar result holds for the semigroup TX of all full transformations on X [7] and the symmetric inverse semigroup IX of all partial one-to-one transformations on X [1]. In the present paper, for various kinds of subgroups G of Sn, we study the induced submonoids M(G)ofCn. In Section 2 we provide definitions, in Section 3 Received by the editors April 12, 2002 and, in revised form, June 5, 2002. 2000 Mathematics Subject Classification. Primary 20M20, 20M15. Key words and phrases. Factorizable monoids, symmetric inverse semigroups, class equation, conjugacy classes, permutation groups, transformation semigroups. c 2003 American Mathematical Society 3345 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3346 S. LIPSCOMB AND J. KONIECZNY we consider orders jM(G)j, and in Section 4, using the class equation for finite monoids, we investigate conjugacy classes. 2. Basic definitions Let M be a monoid with the identity 1. Then u 2 M is a unit when v 2 M exists such that uv = vu =1.Ifsuchav exists, it is unique and denoted u−1. The subset G ⊆ M of all units is a subgroup of M, called the group of units of M. For example, Sn is the group of units of M(Sn)=Cn. The monoid Cn is of particular importance because every finite inverse monoid is isomorphic to a submonoid of Cn for some n [2, Theorem 5.1.7]. The elements of Cn are partial one-to-one transformations on N, they are sometimes called charts [6], and the domain and range of the chart α are denoted, respectively, dα and rα. An element b of a monoid M is called an inverse of a 2 M if a = aba and b = bab.Wheneacha 2 M has exactly one inverse, then M is an inverse monoid. For example, Cn is an inverse monoid since the required unique inverse of α 2 Cn is α−1. Every inverse monoid M has a natural partial order given by a ≤ b whenever an idempotent e 2 M exists such that a = eb [2, p. 152]. An inverse monoid M is factorizable when each a 2 M is \under" a unit u 2 M, i.e., a ≤ u [4]. In other words, an inverse monoid M is factorizable if every element of M can be factored as a product of an idempotent and a unit. For example, Cn is factorizable: For each idempotent " 2 Cn and each x 2 N,eitherx" = x or x" is not defined, i.e., " is completely determined by its domain. So for A ⊆ N, "A will denote the idempotent 2 2 in Cn with domain A. Next, for a unit σ Sn, the element "Aσ Cn is a restriction of σ to A. Indeed, every α 2 Cn is a restriction of some unit σ 2 Sn to A = dα.It follows that Cn is a factorizable inverse monoid. We shall use \G ≤ Sn"toindicatethatG is a subgroup of Sn.Itiseasytosee that the induced monoid M(G) is an inverse submonoid of Cn. And since M(G) consists of all elements of the form εσ where " 2 Cn is idempotent and σ 2 G,it is factorizable. In fact, M(G) is the largest (with respect to inclusion) factorizable submonoid of Cn that has G as its group of units. ≤ ⊆ 2 For G Sn and A N,thestabilizer GA of A consists of all σ G such that 2 f g xσ = x for every x A.IfA = x ,wewriteGA = Gx and call Gx the stabilizer of ≤ x. It is clear that GA is a subgroup of G. A subgroup G Sn is balanced if for all ⊆ j j j j subsets A; B N of the same cardinality, we have GA = GB . For example, Sn and its alternating subgroup An are balanced, while the cyclic subgroup G = h(1 2)i generated by the 2-cycle (12) in S4 is not balanced. A group G ≤ Sn is transitive if for all x; y 2 N,aσ 2 G exists such that y = xσ; semiregular if Gx = f1g for each x 2 N;andregular if it is both semiregular and transitive. Note that every semiregular group G is balanced since GA = f1g for every non-empty A ⊆ N. For abstract monoids M,wesayb 2 M is conjugate to a 2 M if b = u−1au for some unit u 2 M. The subset [a] of all elements of M conjugate to a is the conjugacy class of a;andf [a]:a 2 Mg is a partition of M. ⊆ M f 2 g For each set A N,letRA = "Aσ : σ G denote the set of all elements in M R M = M(G) that have domain A.(ThesetRA is a Green's -class of M;see[2, ; M f g p. 45] for the definition of Green's relations.) Note that if A = ,thenRA = 0 , where 0 is the empty (or zero) transformation on N. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use CLASS EQUATION AND COUNTING IN FACTORIZABLE MONOIDS 3347 3. Order of M(G) The next lemma is crucial in proving the counting theorems of this section. Lemma 3.1. Let G ≤ Sn and M = M(G).ForeachA ⊆ N, jGj j M j RA = : jGAj M ! Proof. Define f : RA G=GA by f("Aσ)=GAσ.Thenf is well-defined and one-to-one since for all σ, ρ 2 G, −1 −1 "Aσ = "Aρ , "A = "Aρσ , ρσ 2 GA , ρ 2 GAσ , GAσ = GAρ. j M j j j j j Since the function f is clearly onto, it follows that RA is the number G = GA of right cosets of GA in G. The next three theorems give, respectively, formulas for orders of monoids in- duced by balanced, semiregular, and transitive abelian groups. Let k 2f0; 1;::: ;ng. Then C(n; k) denotes \n choose k" (the number of k-element subsets of N), and for a balanced group G ≤ Sn, mk denotes jGf1;::: ;kgj.Thatis,mk is the size of the ⊆ j j stabilizer GA for any k-element set A N.NotethatG; = G and so m0 = G . Theorem 3.2. Let G ≤ Sn be balanced and M = M(G).Then Xn C(n; k) jMj = jGj : mk k=0 ⊆ M M Proof. When A; B N are distinct, RA and RB are disjoint (the domain of any 2 M 2f g "Aσ RA is A). So for each k 0; 1;::: ;n ,thereareC(n; k) pairwise disjoint M j j sets RA with A = k. But since G is balanced, Lemma 3.1 shows that each set M j j j j j j j j j j RA with A = k has G = GA = G =mk elements. Thus, M has C(n; k)( G =mk) elements with domains of size k. The result follows. To illustrate Theorem 3.2, first suppose G = Sn.ThenM = Cn and mk = (n − k)! for every k, showing that Xn C(n; k) jC j = n! : n (n − k)! k=0 c Second, suppose G = An.ThenM = An, the alternating monoid. (The monoid c An, which is a natural extension of the alternating group, was discovered by the ≥ − (n−k)! first author [5].) For G = An,wehavemk =1(ifk n 1) and mk = 2 (if k ≤ n − 2).
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