Relative Ranks of Some Partial Transformation Semigroups
Turkish Journal of Mathematics Turk J Math (2019) 43: 2218 – 2225 http://journals.tubitak.gov.tr/math/ © TÜBİTAK Research Article doi:10.3906/mat-1902-23
Relative ranks of some partial transformation semigroups
Ebru YİĞİT1, Gonca AYIK2,,∗ Hayrullah AYIK2 1Department of Informatics, Faculty of Engineering, Yalova University, Yalova, Turkey 2Department of Mathematics, Faculty of Arts and Sciences, Çukurova University, Adana, Turkey
Received: 05.02.2019 • Accepted/Published Online: 16.07.2019 • Final Version: 28.09.2019
Abstract: Let Pn , Tn , In , and Sn be the partial transformation semigroup, the (full) transformation semigroup, the symmetric inverse semigroup, and the symmetric group on Xn = {1, . . . , n}, respectively. For 1 ≤ r ≤ n − 1, let PKn,r be the subsemigroup consisting α ∈ Pn such that |imα| ≤ r and let SPKn,r = PKn,r \ Tn . In this paper, we first examine the subsemigroup In,r = In ∪ PKn,r and we find the necessary and sufficient conditions for any nonempty subset of PKn,r to be a (minimal) relative generating set of the subsemigroup In,r modulo In . Then we examine the subsemigroups PIn,r = SIn ∪ PKn,r and SIn,r = SIn ∪ SPKn,r for 1 ≤ r ≤ n − 1 where SIn = In \ Sn and compute their relative rank.
Key words: (Partial) transformation semigroup, symmetric inverse semigroup, symmetric group, (minimal) generating set, relative rank
1. Introduction
The partial transformation semigroup PX , the (full) transformation semigroup TX and the symmetric inverse semigroup IX on a set X have been extensively studied over the last sixty years, both in the finite and in the infinite cases. Among recent contributions are[1–6, 13, 16]. Here we are concerned solely with the case where { } X = Xn = 1, . . . , n , and we denote the semigroups PXn , TXn , and IXn , by Pn , Tn , and In , respectively.
Moreover, we denote the subsemigroup In \ Sn by SIn where Sn is the symmetric group on Xn .
It is well known that In is an inverse semigroup and every finite inverse semigroup S is embeddable in In , the analog of Cayley’s theorem for finite groups. Hence, as emphasized in[1], the importance of In to inverse semigroup theory is similar to that of the symmetric group Sn to group theory. Moreover, Gomes and Howie remarked in [11] that very little has been written on the symmetric inverse semigroups. Despite the appearance of the books of Lipscomb [18], and Ganyushkin and Mazorchuk [8], as well as a handful of papers
(for example, [10]), the study of In is still in its infancy compared to that of Tn . 2 An element α of Pn is called an idempotent if α = α. We denote the set of all idempotents in any subset U of any semigroup by E(U). Let S be a semigroup and let A be a nonempty subset of S . Then the subsemigroup generated by A, that is the smallest subsemigroup of S containing A, is denoted by ⟨A⟩. If a semigroup S has a finite subset A such that S = ⟨A⟩, then S is called a finitely generated semigroup. The rank of a finitely generated semigroup S is defined by rank(S) = min{ |A| : ⟨A⟩ = S }. For a fixed subset G of
∗Correspondence: [email protected] 2010 AMS Mathematics Subject Classification: 20M20
2218 This work is licensed under a Creative Commons Attribution 4.0 International License. YİĞİT et al./Turk J Math a semigroup S , if there exists a subset A of S such that ⟨A ∪ G⟩ = S , then A is called a relative generating set of S modulo G. Then the relative rank of a finitely generated semigroup S modulo G is defined by
rerank(S : G) = min{|A| : ⟨A ∪ G⟩ = S}.
For 1 ≤ r ≤ n, let
Kn,r = {α ∈ Tn : |im(α)| ≤ r}, Tn,r = Sn ∪ Kn,r,
PKn,r = {α ∈ Pn : |im(α)| ≤ r}, PTn,r = Sn ∪ PKn,r,
SPKn,r = PKn,r \ Tn = PKn,r \ Kn,r,An,r = An ∪ Kn,r,
PAn,r = An ∪ PKn,r, In,r = In ∪ PKn,r,
SIn,r = SIn ∪ SPKn,r and PIn,r = SIn ∪ PKn,r, where An denotes the alternating group on Xn .
Howie and McFadden proved in [15] that the rank of Kn,r is S(n, r), the Stirling number of the second kind, for 2 ≤ r ≤ n − 1. Recall that the Stirling number S(n, r) of the second kind is defined by
S(n, 1) = S(n, n) = 1 and S(n, r) = S(n − 1, r − 1) + r · S(n − 1, r) for 2 ≤ r ≤ n−1. Moreover, Garba proved in [9] that the rank of the subsemigroup PKn,r of Pn is S(n+1, r+1) for 2 ≤ r ≤ n − 1. + For n, r ∈ Z with r ≤ n, let Pr(n) be the set of all integer solutions of the equation
x1 + x2 + ··· + xr = n with x1 ≥ x2 ≥ · · · ≥ xr ≥ 1, and let pr(n) = |Pr(n)|. If an r-tuple (n1, n2, . . . , nr) is a solution of the equation given above, then it is called a partition of n with r terms (see [12]). Ayık et al. developed a notation for certain primitive elements of Tn , called path-cycle, and described an algorithm to decompose an arbitrary transformation α in Tn into a product of path-cycles in [2]. In addition, they used these techniques to obtain some informations about generators of
Tn , and proved that, rerank(Tn,r : Sn) = pr(n) for 1 ≤ r ≤ n − 1 (see also [17, Theorem 8]).
In [19], we obtained the necessary and sufficient conditions for any nonempty subset U of Kn,r (PKn,r) to be a (minimal) relative generating set of Tn,r (PTn,r) modulo Sn for 1 ≤ r ≤ n − 1. Then we concluded the same result in [2, 17] that rerank(Tn,r : Sn) = pr(n) and, we obtained the new result
n∑−r rerank(PTn,r : Sn) = pr(n − s) s=0 for 1 ≤ r ≤ n − 1. Moreover, we showed that
n∑−r rerank(An,r : An) = pr(n) and rerank(PAn,r : An) = pr(n − s) s=0 for each 1 ≤ r ≤ n − 1. In this paper, we first find the necessary and sufficient conditions for any nonempty subset U of
PKn,r to be a (minimal) relative generating set of In,r (respectively PIn,r ) modulo In (respectively SIn )
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for 1 ≤ r ≤ n − 1. Moreover, we find the necessary and sufficient conditions for any subset U of SPKn,r to be a (minimal) relative generating set of SIn,r modulo SIn . Then we conclude that
rerank(In,r : In) = pr(n),
rerank(SIn,r : SIn) = pr(n − 1) and
rerank(PIn,r : SIn) = S(n, r) for 1 ≤ r ≤ n − 1.
2. Preliminaries
The height and the kernel of any partial transformation α ∈ Pn are defined by h(α) = |im(α)| and
ker(α) = {(x, y) ∈ Xn × Xn : either x, y ∈ dom(α) and xα = yα or x, y∈ / dom(α)}, respectively. For any α, β ∈ Pn (also α, β ∈ Tn ), recall that dom(αβ) ⊆ dom(α), im(αβ) ⊆ im(β), ker(α) ⊆ ker(αβ), and that
(α, β) ∈ D ⇔ h(α) = h(β) (α, β) ∈ H ⇔ im(α) = im(β), ker(α) = ker(β) and dom(α) = dom(β) where the equivalences D and H denote Green’s relations (see, for examples [14] and [8, Theorem 4.5.1]). For ≤ ≤ D T 1 r n, we denote that Green’s -class, consists of all elements in Tn (respectively Pn ) of height r, by Dr P (respectively Dr ). If the implied semigroup is clear from the context, we will use the simpler notation Dr . D T It is shown in [7] that Green’s -class Dr is generated by its idempotents, and so it follows from [15, ⟨ T ⟩ ≤ ≤ − D Lemma 4] that Kn,r = E(Dr ) for 2 r n 1. It is also shown in [9] that a subset A of Green’s -class P P ⊆ ⟨ ⟩ ≤ ≤ − Dr is a generating set of PKn,r if and only if E(Dr ) A for 2 r n 1. Therefore, to show a subset P P ⊆ ⟨ ⟩ A of Dr is a generating set of PKn,r , it is enough to prove Dr A .
For a given nonempty set X and a positive integer r where 1 ≤ r ≤ |X|, let A1,...,Ar be a collection of nonempty disjoint subsets of X . Then ξ = {A ,...,A } is called a partition of X (with r terms) if ∪ 1 r r { } | | ≥ · · · ≥ | | X = i=1 Ai . A partition A1,...,Ar of X is called an ordered partition if A1 Ar , and is P denoted by (A1,...,Ar). For 1 ≤ r ≤ n, it is clear that α ∈ D if and only if there exists a unique partition ∪ r { } r+1 × \ A1,...,Ar of dom(α) such that ker(α) = i=1 (Ai Ai) where Ar+1 = Xn dom(α) = cdom(α); or equivalently, there exists a unique subset {a1, . . . , ar} of Xn with cardinality r, such that im(α) = {a1, . . . , ar}.
Without loss of generality suppose that Aiα = ai for each 1 ≤ i ≤ r, and so α can be written in the following tabular form: ( )( ( ) ) ··· ··· A1 Ar Ar+1 A1 Ar ∈ T α = and α = if α Dr . a1 ··· ar − a1 ··· ar
∈ T ∈ For α Dr , written in the above tabular form, there clearly exists a permutation σ Sr such that
(|A1σ|,..., |Arσ|) is a partition of n with r terms. In this case the partition of α is defined by
part(α) = (|A1σ|,..., |Arσ|) .
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∈ P | | | | ≥ ∈ For α Dr if cdom(α) = Ar+1 = s 1, similarly there exists a permutation σ Sr such that
(|A1σ|,..., |Arσ|) is a partition of n − s with r terms. In this case, the co-partition of α is defined by
copart(α) = ( |A1σ|,..., |Arσ| : |Ar+1| ) .
If |cdom(α)| = |Ar+1| = s = 0, then for convenience the copartition of α is defined by copart(α) = part(α). ≤ ≤ − P T ∈ P From now on, we consider the case 1 r n 1, since Dn = Dn = Sn . We also assume that α Dr is in the above tabular form unless stated otherwise. First, recall Proposition 1 and Lemma 2 in [19]:
Proposition 2.1 For 1 ≤ r ≤ n−1, let α, β ∈ Dr . Then αβ ∈ Dr if and only if ker(αβ) = ker(α). Moreover,
αβ ∈ Dr implies cdom(αβ) = cdom(α).
Lemma 2.2 For 1 ≤ r ≤ n−1, let α, β ∈ Dr . Then copart(α) = copart(β) if and only if there exist λ, µ ∈ Sn such that α = λβµ. In particular, for α, β ∈ Tn , part(α) = part(β) if and only if there exist λ, µ ∈ Sn such that α = λβµ.
Next we state and prove the following similar lemma which will be used throughout this paper:
≤ ≤ − ∈ P ≥ Lemma 2.3 For 1 r n 1, let α Dr and let copart(α) = (n1, n2, . . . , nr : s) with s 1.
∈ T ∈ (i) For any β Dr with part(β) = (n1 + s, n2, . . . , nr), there exist λ, µ SIn such that α = λβµ.
∈ P − ≤ ≤ ∈ (ii) For any β Dr with copart(β) = (n1 + s k, n2, . . . , nr : k) where 1 k s, there exist λ, µ SIn such that α = λβµ.
∈ P | | | | ∈ (iii) For any β Dr , copart(α) = copart(β) if and only if dom(α) = dom(β) and there exist λ, µ SIn such that α = λβµ.
Proof Without loss of generality, let ( ) A ··· A A α = 1 r r+1 a1 ··· ar − where |Ai| = ni for 1 ≤ i ≤ r and |Ar+1| = s ≥ 1. ∈ P (i)-(ii) For any β Dr without loss of generality let ( ) B ··· B B β = 1 r r+1 b1 ··· br − where { n1 + s if |Br+1| = 0 |B1| = and |Bi| = ni n1 + s − k if 1 ≤ |Br+1| = k ≤ s for each 2 ≤ i ≤ r. Then it is clear that there exists λ ∈ SIn such that A1λ ⊆ B1 , Aiλ = Bi for each 2 ≤ i ≤ r and cdom(λ) = cdom(α). Moreover, consider the partial injective transformation µ : im(β) −→ im(α) in SIn defined by biµ = ai for each 1 ≤ i ≤ r. Since Ai(λβµ) = ai for each 1 ≤ i ≤ r and cdom(λβµ) = Ar+1 , we have α = λβµ, as required.
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(iii)(⇒) Suppose that copart(α) = copart(β). Then, from (ii), it follows that k = s.
(⇐) Suppose that |dom(α)| = |dom(β)| and that there exist λ, µ ∈ SIn such that α = λβµ. Then dom(α) ⊆ dom(λ) and (dom(α))λ ⊆ dom(β), and so (dom(α))λ = dom(β). Since α, β ∈ DP , it follows that ∪ r ⊆ r im(β) dom(µ), and so (im(β))µ = im(α). Thus, it is easy to see that dom(β) = i=1 Aiλ and that ( ) A1λ ··· Arλ Br+1 β = −1 −1 , a1µ ··· arµ − where Br+1 = cdom(β). Therefore, since |Ai| = |Aiλ| for each 1 ≤ i ≤ r, we have copart(α) = copart(β), as required. 2
For 1 ≤ r ≤ n − 1, let α, β ∈ In with h(α) = h(β) = r. Then it is clear that h(αβ) = r if and only if im(α) = dom(β).
3. Relative ranks
Notice that, since In,r \ In = PKn,r is an ideal of In,r , any generating set of In,r must contain a generating ≤ ≤ − ⊆ ⊆ P ∩ set of In for 1 r n 1. Thus, if W In,r is a generating set of In,r , then there exist U Dr W and
V ⊆ In ∩ W such that In = ⟨V ⟩ and In,r = ⟨U ∪ V ⟩ = ⟨U ∪ In⟩. Therefore, any minimal relative generating P ≤ ≤ − set of In,r modulo In must be a subset of Dr for 1 r n 1.
Theorem 3.1 Let 1 ≤ r ≤ n − 1 and U ⊆ PKn,r . Then In,r = ⟨U ∪ In⟩ if and only if, for each partition ∈ ∈ ∩ T p = (n1, . . . , nr) Pr(n), there exists β U Dr such that part(β) = p.
Proof (⇐) Let 1 ≤ r ≤ n − 1. For each partition p = (n1, . . . , nr) ∈ Pr(n), we fix an arbitrary element ∈ ∩ T { ∈ ∈ } βp U Dr with part(βp) = p. Then we denote the set of all these fixed elements by V = βp U : p Pr(n) . ∈ P ∈ ∈ ∈ For any element α Dr either α Kn,r or α SPKn,r . If α Kn,r with copart(α) = part(α) =
(n1, . . . , nr) = p, then there exists βp ∈ V such that part(βp) = p, and so it follows from Lemma 2.2 that there exist λ, µ ∈ Sn ⊆ In such that α = λβpµ. Now suppose that α ∈ SPKn,r with copart(α) = (n1, n2, . . . , nr : s) where s ≥ 1. Then there exists βp ∈ V such that part(βp) = (n1 + s, n2, . . . , nr) = p and so, it follows from
Lemma 2.3 (i) that there exist λ, µ ∈ SIn ⊆ In such that α = λβpµ. Thus, the set V ∪ In and so, the set ∪ P ∪ U In generates Dr . Therefore, it follows from Lemma 2.6 given in [9] that U In is a generating set of In,r .
(⇒) For 1 ≤ r ≤ n − 1, let U ⊆ PKn,r be a relative generating set of In,r modulo In , that is ⟨ ∪ ⟩ ∈ ∈ T In,r = U In . Then, for an arbitrary partition p Pr(n), consider an arbitrary element β Dr such that part(β) = p. Since In,r = ⟨U ∪ In⟩, β can be written as a product of finitely many elements of U ∪ In . It follows from β ̸∈ In that either β = β1δ or β = α1β1δ for some β1 ∈ U , δ ∈ ⟨U ∪ In⟩ ⊆ In,r and α1 ∈ In \{ε} where ε is the identity permutation on Xn . Now let { β if β = β δ γ = 1 1 α1β1 if β = α1β1δ, and so β = γδ . Since Xn = dom(β) ⊆ dom(γ), it follows that γ ∈ Tn (and α1 ∈ Sn in the second case), and ∈ ≥ ∈ ∈ T so β1 Tn in both cases. Since h(γ) h(β) = r and β1 Kn,r , it follows that γ, β1 Dr . Thus, since
2222 YİĞİT et al./Turk J Math ker(γ) ⊆ ker(β), we have ker(β) = ker(γ) and so,
p = part(β) = part(γ) = part(β1).
∈ ∩ T 2 Therefore, β1 U Dr and part(β1) = p, as required. As an immediate consequence, we have the following corollary:
Corollary 3.2 For each 1 ≤ r ≤ n − 1,
rerank(In,r : In) = pr(n) where pr(n) is the number of partitions of n with r terms.
Notice that, since SIn,r \ SIn = SPKn,r is an ideal of SIn,r , any generating set of SIn,r must contain a generating set of SIn . Similarly, any minimal relative generating set of SIn,r modulo SIn must be a subset P ∩ P \ T ≤ ≤ − of Dr SPKn,r = Dr Dr for each 1 r n 1.
Theorem 3.3 Let 1 ≤ r ≤ n − 2 and U ⊆ SPKn,r . Then SIn,r = ⟨U ∪ SIn⟩ if and only if, for each partition ∈ − ∈ ∩ P (n1, . . . , nr) Pr(n 1), there exists β U Dr such that copart(β) = (n1, . . . , nr : 1).
Proof (⇐) Let 1 ≤ r ≤ n − 2. For each partition p = (n1, . . . , nr) ∈ Pr(n − 1), we fix an arbitrary ∈ ∩ P element βp U Dr with copart(βp) = (n1, . . . , nr : 1). Then we denote the set of all these fixed elements by
V = {βp ∈ U : p ∈ Pr(n − 1)}. ∈ P ∩ ≥ − For any element α Dr SPKn,r , let copart(α) = (n1, . . . , nr : s). Since s 1, we have (n1 + s
1, n2, . . . , nr) = p ∈ Pr(n − 1). Then there exists βp ∈ V such that part(βp) = p, so it follows from Lemma
2.3 (ii) (when k = 1) that there exist λ, µ ∈ SIn such that α = λβpµ. Thus, the set V ∪ SIn , and so the set ∪ P \ T ∪ U SIn generates Dr Dr . It follows from Lemma 2.6 in [9] that U SIn is a generating set of SIn,r .
(⇒) For 1 ≤ r ≤ n − 2, let U ⊆ SPKn,r be a relative generating set of SIn,r modulo SIn . Then, ∈ − ∈ P for an arbitrary partition p = (n1, . . . , nr) Pr(n 1), consider an arbitrary element β Dr such that copart(β) = (n1, . . . , nr : 1). Since SIn,r = ⟨U ∪ SIn⟩, β can be written as a product of finitely many elements of U ∪ SIn . It follows from the fact β ̸∈ SIn that either β = β1δ or β = α1β1δ for some β1 ∈ U , α1 ∈ SIn and δ ∈ SIn,r ∪ {ε} where ε is the identity permutation on Xn . Now let { β if β = β δ γ = 1 1 α1β1 if β = α1β1δ, and so β = γδ . Since |cdom(β)| = 1 and dom(β) ⊆ dom(γ) ≠ Xn , we have dom(β) = dom(γ). Moreover, ≥ ∈ ∈ P ⊆ since h(γ) h(β) = r and β1 SPKn,r , it follows that γ, β1 Dr . Thus, since ker(γ) ker(β) and dom(β) = dom(γ), we have ker(β) = ker(γ), so copart(β) = copart(γ).
Now let γ = β1 , then clearly copart(β) = copart(β1). Otherwise, since α1 ∈ SIn , dom(β) = dom(γ) ⊆ dom(α1) and |cdom(β)| = 1, we have dom(β) = dom(α1) and im(α1) = dom(β1), so
copart(β) = copart(γ) = copart(β1), as claimed. 2 As an immediate consequence, we have the following corollary:
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Corollary 3.4 For each 1 ≤ r ≤ n − 2,
rerank(SIn,r : SIn) = pr(n − 1) where pr(n − 1) is the number of partitions of n − 1 with r terms.
Theorem 3.5 Let 1 ≤ r ≤ n − 1 and U ⊆ PKn,r . Then PIn,r = ⟨U ∪ SIn⟩ if and only if, for each partition { } ∈ ∩ T A1,...,Ar of Xn , there exists β U Dr such that
∪r ker(β) = (Ai × Ai). i=1
Proof (⇒) For 1 ≤ r ≤ n − 1, let U ⊆ PKn,r be a relative generating set of PIn,r modulo SIn . Then, for an arbitrary partition {A ,...,A } of X , consider an arbitrary element β ∈ DT ⊆ K ⊆ PI with ∪ 1 r n r n,r n,r r × ⟨ ∪ ⟩ ∈ ∩ ∈ ∪{ } ker(β) = i=1(Ai Ai). Since PIn,r = U SIn and dom(β) = Xn , there exist β1 U Tn and δ PIn,r ε , where ε is the identity permutation on X , such that β = β δ . Then since β ∈ K , ker(β ) ⊆ ker(β) and n 1 1 n,r ∪ 1 ≥ r × h(β1) h(β) = r, it follows that h(β1) = r, so ker(β) = ker(β1). Therefore, ker(β1) = i=1(Ai Ai) and ∈ ∩ T β1 U Dr , as required.
(⇐) Let 1 ≤ r ≤ n−1. Recall that PKn,r is the disjoint union of SPKn,r and Kn,r . Then consider any ∈ ≥ ∈ T α SPKn,r with copart(α) = (n1, . . . , nr : s). Since s 1, it follows from Lemma 2.3 (i) that for any β Dr ∈ ⟨ T ⟩ with part(β) = (n1 + s, n2, . . . , nr) there exist λ, µ SIn such that α = λβµ. Therefore, since Kn,r = Dr , ⟨ ∪ ⟩ T ⊆ ⟨ ∪ ⟩ to show PIn,r = U SIn it is enough to show that Dr U SIn . T For each partition A = {A ,...,A } of X into r subsets, we fix an arbitrary element βA ∈ U ∩ D ∪ 1 r n r r × such that ker(βA) = i=1(Ai Ai). Then we denote the set of all these fixed elements by
V = {βA ∈ U : A is a partition of Xn into r subsets}. ∪ T r For any α ∈ D , let ker(α) = (Ai × Ai) where A = {A1,...,Ar} is a partition of Xn . Then there r i=1 ∪ ∈ ⊆ ∩ T r × ≤ ≤ exists βA V U Dr such that ker(βA) = i=1(Ai Ai). Let AiβA = bi for 1 i r. Now consider the map δ : im(βA) → im(α) defined by biδ = Aiα for each 1 ≤ i ≤ r. Then it is clear that δ ∈ SIn and that
α = βAδ ∈ ⟨U ∪ SIn⟩, as required. 2 As an immediate consequence, we have the following corollary:
Corollary 3.6 For each 1 ≤ r ≤ n − 1,
rerank(PIn,r : SIn) = S(n, r) where S(n, r) is the Stirling number of the second kind.
Proof The proof follows from the fact that the number of partitions of Xn into r subsets is S(n, r). 2
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