Table Of Content

KROHN–RHODES COMPLEXITY OF BRAUER TYPE SEMIGROUPS 2 1 K.AUINGER 0 2 r Abstract. The Krohn–Rhodes complexity of the Brauer type semi- a groupsBn andAn iscomputed. Inthree-quarterofthecasestheresult M isthe‘expected’one: thecomplexitycoincideswiththe(essential) J- depth of the respective semigroup. The exception (and perhaps the 4 1 most interesting case) is the annular semigroup A2n of even degree in which case the complexity is the J-depth minus 1. For the par- ] tial versions PBn and PAn it is shown that c(PBn) = c(Bn) and R c(PA2n−1) = c(A2n−1) for all n ≥ 1. The computation of c(PA2n) is G left as an open problem. . h t a ItfollowsfromthefamousKrohn–RhodesPrimeDecompositionTheorem m [10] that each finite semigroup S divides an iterated wreath product [ An≀Gn≀An−1≀···≀A1≀G1≀A0 4 v were the Gi are groups and the Ai are aperiodic semigroups. The number 2 n of non-trivial group components of the shortest such iterated product 8 is the group complexity or Krohn–Rhodes complexity of the semigroup S. 9 The question whether this number is algorithmically computable given the 4 . semigroup S as input is perhaps the most fruitful research problem in 2 finite semigroup theory. To the author’s knowledge, this problem is still 0 2 opendespite thetremendous effort that has been spenton it over theyears. 1 Concerning classes of abstract semigroups, the largest pseudovariety : v which contains semigroups of arbitrarily high complexity and for which the Xi complexity of each member can be computed is LG(cid:13)m A — this includes DS and thus completely regular semigroups (unions of groups). Another r a result is that the “complexity-1” pseudovarieties A ∗ G and G ∗A have 2 decidable membership (the latter being contained in LG (cid:13)m A). On the other hand, the complexity of many naturally occurring individual and concrete semigroups is known: these include the semigroup of all transformations of a finite set and the semigroup of all endomorphisms of a finite 2010 Mathematics Subject Classification. 20M07, 20M20, 20M17. Keywordsandphrases. Krohn–Rhodescomplexity,Brauertypesemigroup,pseudovariety of finite semigroups. 1 2 K.AUINGER vector-space [16], as well as the semigroup of all binary relations on a finite set [17]. More recently, Kambites [9] calculated the complexity of the semigroup of all upper triangular matrices over a finite field. The present paper intends to contribute to the latter kind of results. Indeed, we shall present a calculation of the complexity of the Brauer semigroup B and n the annular semigroup A (these occur originally in representation theory n of associative algebras but have recently attracted considerable attention among semigroup theorists). It turns out that the cases of B and A n 2n+1 can be treated in a straightforward fashion by the use of arguments that apply to transformation semigroups and linear semigroups. The case of the annular semigroup A of even degree is somehow different. Although the 2n problem can be solved by use of the machinery developed by the Rhodes school, the solution requires quite a bit of care and is much less obvious. Actually, the author was not able to compute the complexity c(A ) di- 2n rectly. The strategy is rather to look at a certain natural subsemigroup EA first and calculate the complexity of this subsemigroup. In a second 2n step it is then shown that the complexity of the full semigroup A does 2n not exceed the complexity of the ‘even’ subsemigroup EA . It should be 2n mentioned that none of the semigroups B or A is contained in LG(cid:13)m A n n except for n = 1. Thepaperisorganized asfollows. Insection 1wecollect allpreliminaries on Brauer and annular semigroups as well as the basics of Krohn–Rhodes complexity needed in the sequel. In section 2 the complexity of the Brauer semigroup is computed to be c(B ) = ⌊n⌋ which is exactly the (essential) n 2 J-depth. It is also shown that the partial Brauer semigroup PBn has the same complexity as the ‘total’ counterpart B . In section 3 we first treat n the annular semigroup of even degree and show that c(A ) = n −1. This n 2 is the difficult case and is treated with the help of a certain subsemigroup, theeven annular semigroup EA . Afterwardstheodddegreecaseistreated n which is again, in a sense, standard. Finally, some remarks on the partial versions PA are given. In the oddcase, the complexity is the same as that n of their total counterparts while the computation of the complexity in the even case is left as an open problem. Throughout the paper, all semigroups are assumed to be finite. For background information on (finite) semigroups the reader is referred to the monographs by Almeida [1] and Rhodes and Steinberg [18]. 3 1. Preliminaries 1.1. Brauer type semigroups. Here we present the basic definitions and results concerning Brauer type semigroups. For each positive integer n we are going to define: • the partition semigroup C , n • the Brauer semigroup B , n • the partial Brauer semigroup PB , n • the Jones semigroup J , n • the annular semigroup A , n • the partial annular semigroup PA . n The semigroups C , B , A and J arise as vector space bases of certain n n n n associative algebras which are relevant in representation theory [6, 8, 7]. The semigroup structure and related questions for the above-mentioned semigroups have been studied by several authors, see, for example, [3, 4, 11, 13, 14, 15]. We start with the definition of C . For each positive integer n let n ′ ′ ′ ′′ ′′ ′′ [n]= {1,...,n}, [n] = {1,...,n}, [n] ={1 ,...,n } be three pairwise disjoint copies of the set of the first n positive integers and put ′ [n]= [n]∪[n]. The base set of the partition semigroup C is the set of all partitions of the n f set [n]; throughout, we consider a partition of a set and the corresponding equivalence relation onthatset as two differentviews ofthesamethingand withfout further mention we freely switch between these views, whenever it seems to be convenient. For ξ,η ∈ C , the product ξη is defined (and n computed) in four steps [20]: ′ ′ ′ ′′ (1) Consider the -analogue of η: that is, define η on [n] ∪[n] by ′ ′ ′ x η y :⇔ x η y for all x,y ∈ [n]. ′′ ′ (2) Lethξ,ηibetheequivalencerelation on[n]∪[n] generatedbyξ∪η, f that is, set hξ,ηi := (ξ∪η′)t where t denotes the transitive closure. ′ (3) Forget all elements having a single primfe : that is, set ◦ hξ,ηi := hξ,ηi|[n]∪[n]′′. (4) Replace double primes with single primes to obtain the product ξη: that is, set ◦ x ξη y :⇔ f(x)hξ,ηi f(y) for all x,y ∈ [n] f 4 K.AUINGER ′′ where f :[n] → [n]∪[n] is the bijection ′ ′′ x 7→ x,x 7→ x for all x ∈ [n]. f This multiplication is associative making C a semigroup with identity 1 n where ′ 1= {{k,k } |k ∈ [n]}. The group of units of C is the symmetric group S (acting on [n] on the n n right) with canonical embedding S ֒→ C given by n n ′ σ 7→ {{k,(kσ) } |k ∈ [n]} for all σ ∈ S . n Moregenerally, thesemigroup of all(total) transformations T of [n]acting n on the right is also naturally embedded in C by n φ7→ {{k′}∪kφ−1 |k ∈ [n]}. (1) ′ If k is not in the image of φ then {k } forms by definition a singleton class. Theequivalence classes of some ξ ∈ C are usually referred to as blocks; the n rank rkξ is the number of blocks of ξ whose intersection with [n] as well as ′ with [n] is not empty — this coincides with the usual notion of rank of a mapping on [n] in case ξ is in the image of the embedding (1). It is known that the rank characterizes the D-relation in Cn [14, 11]: for any ξ,η ∈ Cn, one has ξ D η if and only if rkξ = rkη. ThesemigroupC admitsanaturalinverseinvolution makingitaregular n ∗ ∗-semigroup: considerfirstthepermutation on[n]thatswapsprimedwith unprimed elements, that is, set ∗ ′ ′ ∗ f k = k , (k ) = k for all k ∈ [n]. Then define, for ξ ∈ C , n ∗ ∗ ∗ x ξ y :⇔ x ξ y for all x,y ∈ [n]. ∗ That is, ξ is obtained from ξ by interchanging in ξ the primed with the f unprimed elements. It is easy to see that ∗∗ ∗ ∗ ∗ ∗ ξ = ξ, (ξη) = η ξ and ξξ ξ = ξ for all ξ,η ∈ C . (2) n ∗ The elements of the form ξξ are called projections. They are idempotents (as one readily sees from the last equality in (2)). We note that in the ∗ groupH -class of any projection, theinvolution coincides with the inverse operation in that group. The Brauer semigroup B can be conveniently defined as subsemigroup n of C : namely, B consists of all elements of C all of whose blocks have n n n size 2. It is useful to think of the elements of B in terms of diagrams. n These are pictures like the one in Figure 1. 5 ′ ′ ′ ′ ′ ′ ′ 1 2 3 4 5 6 7 1 2 3 4 5 6 7 Figure 1. A diagram in B 7 The partial Brauer semigroup PB consists of all elements of C all of n n whose blocks have size at most 2. The elements of PB admit a similar in- n terpretationintermsofdiagrams. BothsemigroupsB andPB areclosed n n ∗ under . In both types of semigroups, the group H -class of a projection ∗ ξξ of rank t is isomorphic (as a regular ∗-semigroup) with the symmetric group S . Let ξ ∈ B be of rank t and let t n ′ ′ {k ,l },...,{k ,l }, for some k ,l ∈ [n], 1 1 t t i i ′ be the blocks of ξ which contain an element of [n] and of [n]. Then ′ ′ {k ,...,k } and {l ,...,l } is the domain domξ respectively range ranξ 1 t 1 t ′ of ξ. For any projection ε we obviously have ranε= (domε). The Jones semigroup (also called Temperly–Lieb semigroup)1 J is the n subsemigroup of B consisting of all diagrams that can be drawn in the n plane within a rectangle (as in Figure 1) in a way such that any two of its lines have empty intersection. These diagrams are called planar. It is well-known and easy to see that J is aperiodic [12]. n Next we define the annular semigroup A [8]. It will also be realized n as a certain subsemigroup of the Brauer semigroup. For this purpose it is convenient to first represent the elements of B as annular diagrams. n Consider an annulus A in the complex plane, say A = {z |1 < |z| < 2} and identify the elements of [n] with certain points of the boundary of A via 2πi(k−1) ′ 2πi(k−1) k 7→ e n and k 7→ 2e n for all k ∈[n]. f For ξ ∈ B take a copy of A and link any x,y ∈ [n] with {x,y} ∈ ξ by a n path (called string) running entirely in A (except for its endpoints). For example, the element ξ ∈B given by f 4 ′ ′ ′ ′ ξ = {{1,1 },{2,4},{2,3 },{3,4}} is then represented by the annular diagram in Figure 2. Paths representing 1Following [12], we use theterm Jones semigroup. 6 K.AUINGER ′ 2 2 3′ 3 1 1′ 4 ′ 4 Figure 2. Annular diagram representation of a member of B 4 ′ ′ ′ blocks of the form {x,y } [{x,y} and {x,y }, respectively] for some x,y ∈ [n] are called through strings [inner and outer strings, respectively]. The annular semigroup A by definition consists of all elements of B that n n have a representation as an annular diagram any two of whose strings have empty intersection. One can compose annular diagrams in an obvious way, modelling the multiplication in B — from this it follows that A is closed n n ∗ under the multiplication of B . Clearly, A is closed under , as well. n n AnalogouslytothepartialBrauersemigroupPB ,onecanalsodefinethe n partial annular semigroup PA by considering all elements of PB which n n admit a representation by an annular diagram in which any two distinct ∗ strings have empty intersection. Again each PA is closed under . n Finally, we fix the following notation: if the semigroup M happens to be × a monoid then its group of units is denoted by M while the singular part × of M, that is, the subsemigroup of all non-invertible elements M\M is denoted by SingM. 1.2. Krohn–Rhodes complexity. Here we present the basics of Krohn– Rhodescomplexity neededin thesequel. Acomprehensivetreatment of the subject can be found in Part II of the monograph [18]. Throughout, the complexity of a semigroup S is denoted by c(S). A J-class of a semigroup is essential if it contains a non-trivial subgroup. The depth of a semigroup is the length n of the longest chain J1 > J2 > ··· > Jn of essential J- classes. The complexity c(S) of a semigroup S can never exceed its depth [18, Theorem 4.9.15]. Lemma 1. For each semigroup S and for each ideal I of S the inequality c(S) ≤ c(I)+c(S/I) holds. 7 ThelatterstatementisusuallyknownastheIdeal Theorem [18,Theorem 4.9.17]. The next result (due to Allen and Rhodes) can be also found as Proposition 4.12.20 in [18]. Lemma 2. Let S be a semigroup and let e be an idempotent of S; then c(SeS) = c(eSe). The following result [18, Proposition 4.12.23] is useful for computing the complexity of the full transformation semigroup T and the full linear n semigroup M (F ) over a finite field. In our situation it is helpful for the n q Brauer semigroup B and the annular semigroup A of odd degree. n 2n+1 Proposition 3. Suppose that S is a monoid with non-trivial group of units G such that S = hG,ei for some idempotent e ∈/ G and SeS ⊆ hE(S)i. Then c(S) = c(eSe)+1. According to [18, Definition 4.12.11] a semigroup S is a T -semigroup if 1 there exists an L-chain s1 ≤L s2 ≤L ··· ≤L sn of elements of S such that S is generated by s1,...,sn. The type II subsemigroup KG(S) of a semigroup S consists of all elements of S that relate to 1 under every relational morphism from S to a group G. Ash’s famous theorem [2] (verifying the Rhodes type II conjecture) states that KG(S) is the smallest subsemigroup of S that contains all idempotents and is closed under weak conjugation. The combination of Theorems 4.12.14 and 4.12.8 in [18] yields the final preliminary result. Proposition 4. For each T -semigroup S which is not aperiodic, the in- 1 equality c(KG(S)) < c(S) holds. 2. The Brauer semigroup B n It should be mentioned that the full partition semigroup C has com- n plexity n − 1 for each n. Indeed, it has n − 1 essential J-classes hence c(C ) can be at most n − 1. On the other hand, the full transformation n semigroup T on n letters embeds into C and it is a classical result [18, n n Theorem 4.12.31] that c(T ) = n−1. So c(C ) has to be at least n−1. Of n n course, the Jones semigroup J has complexity 0 for each n. n Let us next consider the Brauer semigroup B . Note that B as well as n 2n B2n+1 have n essential J-classes. For each pair i< j with i,j ∈ [n] define the diagram γ as follows: ij ′ ′ ′ γ := {{i,j},{i ,j },{k,k } |k 6= i,j}. (3) ij Each γ is a projection of rank n−2. Proposition 2 in [13] tells us that the ij singular part of B is generated by the projections γ : n ij SingB = hγ | 1 ≤ i< j ≤ ni. (4) n ij 8 K.AUINGER Recall that the group of units of B is the symmetric group on n letters, n denoted S . Another result to be essential is that B is generated by its n n group of units together with some γ [11, Theorem 3.1]: ij B = hS ,γ i for all i< j. (5) n n ij Since BnγijBn = SingBn ⊆ hE(Bn)i and Bn−2 ∼= γijBnγij Proposition 3 implies: Proposition 5. The equality c(Bn)= c(Bn−2)+1 holds for each n ≥ 2. Taking into account that c(B ) = 0 and c(B ) = 1 we obtain already 1 2 the main result of this section: Theorem 6. The equalities c(B ) = c(B ) = n hold for each positive 2n 2n+1 integer n. (n−2) For the partial analogue PB let n ≥ 2 and denote by PB the n n ideal of PB consisting of all elements of rank at most n−2. The Rees n (n−2) quotient PB /PB is an inverse semigroup and therefore has com- n n plexity 1. Since PBn(n−2) = PBnγijPBn and γijPBnγij ∼= PBn−2 and by use of the Ideal Theorem (Lemma 1) and Lemma 2 it follows that c(PBn) ≤ c(PBn−2)+1 holds for each n ≥ 2. In other words, going from PBn−2 to PBn, the complexity increases by at most 1. Since c(PB1) = 0 and c(PB ) = 1 (the former is aperiodic, the latter has only one essential 2 J-class) it follows by induction that c(PB2n) ≤ n and c(PB2n+1) ≤ n for all n. On the other hand, since c(PB ) ≥ c(B ) the reverse inequalities n n also hold. Altogether we have proved: Corollary 7. The equalities c(PB )= c(B ) = ⌊n⌋ hold for each positive n n 2 integer n. 3. The annular semigroup A n 3.1. Even degree. This seems to be the most interesting case. It is not possibleto apply Proposition 3 herebecause SingA is not idempotent gen- n erated (nor contained in the type II subsemigroup). We shall not calculate the complexity of A directly but rather study a certain natural subsemi- n group — the even annular semigroup EA —, calculate the complexity of n the latter and then show that A has no bigger complexity than EA . n n Throughout this subsection let n be even. Let α ∈ A be of rank r and n ′ ′ ′ let a < a <··· < a and b < b < ··· < b be the elements of domα and 1 2 r 1 2 r ranα, respectively. Then the numbers a are alternately even and odd, and i likewise are the numbers b . This is because the nodes strictly between a i i ′ ′ anda aswellasstrictly betweenb andb areentirely involved inouter i+1 i i+1 9 strings respectively inner strings and hence an even number of nodes must ′ ′ ′ be between a and a respectively b and b . A through string {i,j } of i i+1 i i+1 ′ α is even if i−j is even, and otherwise it is odd. Suppose that {a ,b } 1 s+1 is a through string of α. Then, by the definition of A , the other through n ′ strings of α are exactly the strings {a ,b } (where the sum s+i has to i s+i be taken modr). It follows that either all through strings of α are even or all are odd. Define the element α to be even if all of its through strings are even (or equivalently, if α has no odd through string) — note that the even members of A coincide with the oriented diagrams in [8]. All diagrams of n rank 0 are even, by definition. Let α,β ∈ A and suppose that s = −•−−−−• n k l′ is a through string in αβ. By definition of the product in A there exist a n uniquenumber s ≥ 1 and pairwisedistinct u1,v1,u2,...,vs−1,us ∈ [n] such that s is obtained as the concatenation of the strings •−−−−−• •−−−−−• •−−−−−• ··· •−−−−−• •−−−−−• •−−−−−• k u′1 u1 v1 v1′ u′2 us−1 vs−1 vs′−1 u′s us l′ whereu := •−−−−−• is a through stringof α, v := •−−−−−• is a throughstring k u′ us l′ 1 of β, all •−−−−−• are inner strings of β and all •−−−−−• are outer strings of ui vi vi′ u′i+1 α. It is easy to see that for each outer string {i,j} and each inner string ′ ′ {g ,h}ofanyelementγ ofA theinequalitiesi 6≡ j mod2andg 6≡ h mod2 n hold. It follows that u 6≡ v 6≡ u mod2 and therefore u ≡ u mod2 i i i+1 i i+1 for all i whence u ≡ u mod 2. Consequently, s is even if and only if u and 1 s v are both even or both odd while s is odd if and only if exactly one of u and v is even. In particular, the set EA of all even members of A forms a n n submonoidof A . Moreover, since each projection is even, each idempotent n (being the product of two projections) is also even so that EA contains n all idempotents of A . A direct inspection shows that each planar diagram n α∈ J is also even, whence J is a submonoid of EA . n n n Similarly as inAn andJn, Green’s J-relation inEAn is characterized by the rank: two diagrams of EAn are J-related if and only if they have the same rank. The argument is as follows: let ε and η be arbitrary projections ′ ′ ′ of rank t with a < a < ··· < a the domain of ε and b < b < ··· < b 1 2 t 1 2 t the range of η; define γ to be the element having the same inner strings as ′ ′ ε, the same outer strings as η and the through strings {a ,b },...,{a ,b } 1 1 t t in case a ≡ b mod2 while in case a 6≡ b mod2 the through strings of γ 1 1 1 1 ′ ′ ′ ∗ can be chosen to be {a ,b },{a ,b }...,{a ,b }. Then γ ∈ EA , ε = γγ 1 2 2 3 t 1 n ∗ and η = γ γ. 10 K.AUINGER × As far as the group of units EA of EA is concerned, we see that the n n diagram ′ ′ ′ ζ = {{1,2 },{2,3 }},...{n,1}} (6) is odd and so definitely does not belong to EA . On the other hand, n ζ2 = {{1,3′},{2,4′},...,{n−1,1′},{n,2′}} is even whence the group of units EA× is cyclic of order n. More generally, n 2 for each (even) r < n the maximal subgroups of the J-class of all rank-r- elements of EA is cyclic of order r. n 2 We are going to define two actions S and T of Z on A . The action of n S is by automorphisms, but that of T is by translations. For k ∈ Z let S ,T : A → A be defined as follows: αS is the diagram obtained from k k n n k ′ ′ ′ αby replacingeach string{i,j} [respectively {i,j }, {i,j }]by{i+k,j+k} ′ ′ ′ [respectively {i+k,(j+k)}, {(i+k),(j+k)}]; αT is obtained from α by k ′ ′ ′ ′ replacingeachstring{i,j }[respectively {i,j }]by{i,(j+k)}[respectively ′ ′ {(i+k),(j +k)}]. The addition + has to be taken mod n, of course. We call S the shift by k and T the (outer) twist by k. Note that an outer k k twist leaves unchanged all innerstrings. We could similarly definetheinner twist by k but we will not need it. Shifts and twists can be expressed in terms of the unit element ζ defined in (6). Namely, for each α ∈ A and each k ∈ Z the following hold: n αS = ζ−kαζk and αT = αζk. (7) k k For later use we note that αS is even for every k and αT is even for every k k even k if and only if α itself is even. InthefollowingweshallshowthatthesingularpartofEA isidempotent n generated. Inordertosimplifynotation,weset,foreachi ∈ [n],γ := γ , i i,i+1 that is, γ denotes the projection i ′ ′ ′ γ = {{i,i+1},{i ,(i+1)},{k,k } | k 6= i,i+1}. i (Addition has to be taken mod n.) More precisely, we intend to show that SingEA is generated by the projections γ ,...,γ . Set S := hγ ,...,γ i; n 1 n 1 n ∗ thenobviouslyS ⊆SingEA andSisclosedunder . Moreover, Sisclosed n under S±1 and therefore closed under Sk for all k ∈ Z. This is immediate from the fact that the set {γ1,...,γn} is closed under S±1 and that each S is an automorphism. Next, we note that k ′ ′ ′ γn−1···γ1 = {{n−1,n},{1,2},{k,(k+2)} |k = 1,...n−2} =:λ (see Figure 3). The element λ clearly belongs to S, and therefore so does each shifted version λS of λ. Let α be a singular element of A containing the outer k n