COMPLEXITY PSEUDOVARIETIES ARE NOT LOCAL; TYPE II SUBSEMIGROUPS CAN FALL ARBITRARILY IN COMPLEXITY JOHN RHODES AND BENJAMIN STEINBERG Abstract. We prove the following two results announced by Rhodes: The Type II subsemigroup of a finite semigroup can fall arbitrarily in complexity; the complexity pseudovarieties Cn (n ≥ 1) are not local.
1. Introduction and Key Results It is asserted in [5] that there are semigroups (all semigroups in this paper m G) m G (Rhodes unpublished, are finite) of arbitrary complexity in (A 1972). Equivalently, it is asserted that if S is a semigroup, then the Type II subsemigroup [5, 15] KG (S) of S can drop arbitrarily in complexity. In [11] Rhodes constructs examples of semigroups of complexity two and three in m G) m G. The unpublished examples of Rhodes were obtained via (A Kernel Systems [11], iterated matrix semigroup theory [23, 12, 13] and the Presentation Lemma [2]; the proofs were too cumbersome to write up. Similarly it is asserted [22] that the complexity pseudovarieties Cn (n > 1) are not local in the sense of Tilson (Rhodes unpublished). This follows for complexity one and complexity two by the results of [11], but again the general case was simply too messy to publish. Steinberg has since developed a simplification of the Presentation Lemma [19] and some new ideas on working with iterative matrix semigroups [14] (used to prove the complexity pseudovarieties are not finitely based) that allow for a reasonably clean presentation of these results of Rhodes. Let A be the pseudovariety of aperiodic semigroups and G the pseudovariety of groups. Define C0 = A and, inductively, Cn = A ∗ G ∗ Cn−1 ,
where ∗ denotes the semidirect product of pseudovarieties [3]. The KrohnRhodes theorem [7] says that the Cn exhaust the pseudovariety of all semigroups. The complexity [8] of a semigroup S, denoted c(S), is then the least n for which S ∈ Cn . Date: May 6, 2004. 1991 Mathematics Subject Classification. 20M07. Key words and phrases. Complexity, Presentation Lemma, Type II. The second author was supported in part by NSERC and by POCTI approved project POCTI/32817/MAT/2000 in participation with the European Community Fund FEDER. 1
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m W is the pseuIf V, W are pseudovarieties, then the Malcev product V dovariety of all semigroups S with a relational morphism [3] ϕ : S → T ∈ W such that eϕ−1 ∈ V for all idempotents e of T . The Fundamental Lemma of Complexity [10, 21] states that m (G ∗ Cn−1 ) Cn = A
for n > 0. Define KG (S) to be the intersection of 1ϕ−1 over all relational morphisms ϕ : S → G ∈ G; KG (S) is called the Type II subsemigroup of S [15, 5] (or m G if and sometimes the group kernel of S). It is not hard to show S ∈ V only if KG (S) ∈ G [5]. Ash proved [1] that KG (S) is the least subsemigroup of S containing the idempotents, which is closed under weak conjugation [5]; this result was independently obtained by Ribes and Zalesski˘ı [16]; see [5] 0 (S) = S and K n (S) = K n−1 (K (S)). for a detailed history. Let KG G G G Our main result is the following theorem. Theorem 1.1. For each n > 0, there exists a monoid Sn such that 2 c(Sn ) = n + 1 and KG (Sn ) ∈ A.
m G and in particular has complexity 1 (in fact comThus KG (Sn ) ∈ A 1 plexity 2 in the terminology of [6]).
We have the following corollaries that we prove now. Corollary 1.2. There exist monoids of arbitrary complexity in the pseum G) m G. dovariety (A Proof. This is immediate from the theorem and our discussion above.
Corollary 1.3. For each n, there exists a monoid Sn of complexity n + 1 and an onto homomorphism ϕn : Sn G ∈ G such that 1ϕ−1 ∈ A ∗ G. The proof of Theorem 1.1 in fact shows that G can be taken to be an elementary Abelian 2-group. For the following corollaries, we assume familiarity with the Derived Category Theorem [22]. We use the technique of [11, 18]. Corollary 1.4. Cn is not a local in the sense of Tilson for n > 0. m G) m Proof. Let n > 0 and let S be a monoid of complexity n + 2 in (A G. Then there is a relational morphism ϕ : S → G ∈ G with m G ⊆ C1 . 1ϕ−1 ∈ A
Hence the derived category Dϕ is locally in Cn . If Cn were local, then the Derived Category Theorem [22] would imply S ∈ Cn ∗ G ⊆ Cn+1 . But S has complexity n + 2, so Cn cannot be local. Recall that if V is a pseudovariety of monoids (or semigroups), then LV is the pseudovariety of semigroups whose monoid subsemigroups belong to V. A pseudovariety of semigroups is called local by Eilenberg [3] if LV = V.
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Notice that the complexity of a monoid viewed as a semigroup or as a monoid is the same [3]. Corollary 1.5. Let n > 0. Then Cn ( LCn . That is Cn is not local in the sense of Eilenberg. Proof. Let D be the pseudovariety of semigroups whose idempotents form a right zero semigroup, Then D ⊆ A and so Cn ∗ D = Cn for all n ≥ 0. By the Delay Theorem [22], if V is a non-trivial pseudovariety of monoids, then LV = V ∗ D if and only if V is local in the sense of Tilson. Let n > 0. Then since Cn is not local, we have LCn ) Cn ∗ D = Cn ,
as desired.
2. Proof of Theorem 1.1 2.1. Construction of the Sn . The Sn will be constructed iteratively. For now suppose S is a monoid with zero and with non-trivial group of units G. Fix 1 6= g ∈ G. Define F (S, g) as follows. Set A = {a0 , a1 , a2 , a3 , a4 , a5 , a6 }, B = {0, 1, 2, 3, 00 , 20 } and let P : B × A → S be the following matrix (this is the sandwich matrix of the tall fork [11, 2]).
0 1 P = 2 3 00 20
a0 a1 a2 a3 a4 a5 a6 1 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1
We will write bP a for the entry of P in row b, column a. Let S 0 be the quotient of the Rees matrix semigroup M(S, A, B, P ) by the ideal A × 0 × B. Let H = hhi be a cyclic group of order 4 generated by h, written multiplicatively. Let N = HtH = {hi thj | 0 ≤ i, j ≤ 3}. As a set, we define F (S, g) = H ∪ N ∪ S 0 The group of units of F (S, g) will be H. It’s clear how H multiplies against elements of F (S, g) \ S 0 ; we now define how H acts on S 0 ; it suffices to
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consider h. Of course h0 = 0h = 0. Define (a, s, i)h = (a, s, i + 1 mod 4), i = 0, 1, 2, 3 (a, s, i0 )h = (a, s, (i + 2 mod 4)0 ), i = 0, 2 h(ai , s, b) = (ai−1 mod 4 , s, b), i = 0, 1, 2, 3 h(a4 , s, b) = (a5 , s, b) h(a5 , s, b) = (a4 , s, b) h(a6 , s, b) = (a6 , s, b) It’s clear how N multiplies against H. Define N 2 = 0. It remains to define how N multiplies against S 0 . Since we have defined the action of h on S 0 , it suffices to show how t acts on S 0 . Define (2.1) (2.2) (2.3) (2.4)
(a, s, 00 )t = (a, sg, 0) (a, s, 20 )t = (a, s, 2) t(ai , s, b) = (a4 , gs, b), i = 0, 3 t(ai , s, b) = (a5 , s, b), i = 1, 2
and all other products involving t and S 0 to be 0. We remark that the multiplications by g 6= 1 in (2.1) and (2.3) (as opposed to no multiplications in the middle coordinates of (2.2) and (2.4)) are the key to making this construction work. It is straightforward to check associativity (this is just checking the linked equations [9]), c.f. [14]. We choose to identify S with the subsemigroup a0 × S × 0 of F (S, g) (and we call this choice “canonical”). Notice that F (G ∪ 0, g) (recall that G is the group of units of S) is a subsemigroup of F (S, g) and the two “canonical” ways of viewing G ∪ 0 as a subsemigroup of F (S, g) (via F (S, g) and via F (G ∪ 0, g) coincide. Let G0 = hg0 i be a cyclic group of order 4. Let S1 = F (G0 ∪ 0, g0 ) where we take g = g0 . Changing notation, we let G1 = H, g1 = h and N1 = N . Iteratively, we set Sn = F (Sn−1 , gn−1 ) where we set H = Gn with h = gn and Nn = N . Following the conventions established above, we canonically identify Sn−1 with a certain subsemigroup of Sn . The reader is referred to [13, 23] for further examples of such iterated matrix constructions. 2.2. Complexity of Sn . Recall [20] that the depth of a semigroup is the size of the longest chain of J -classes containing non-trivial subgroups. It is straightforward to verify inductively that the depth of Sn is n + 1. Since the Depth Decomposition Theorem [20] states that depth is an upper bound for complexity, we obtain the following. Proposition 2.1. c(Sn ) ≤ n + 1. Our goal is of course to establish the converse. Let V be a pseudovariety. Recall that X ≤ S is called V-pointlike if, for all relational morphisms ϕ : S → T with T ∈ V, there exists t ∈ T such that X ≤ tϕ−1 ; X is called V-idempotent pointlike if one can always choose t to be an idempotent. It
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is easy to see that if X = X 2 and X is V-pointlike, then X is V-idempotent pointlike; we remark that Henckell has proved the converse for certain pseudovarieties, including the complexity pseudovarieties [4]. We use PLV (S) to denote the semigroup of V-pointlike subsets of S. The following proposition is essentially in [17]. It is key to the proof we present here. Proposition 2.2. Suppose S is a semigroup and U ≤ S is a subsemigroup. Suppose that U is W-idempotent pointlike and A ∈ PLV (U ). Then A is m W-pointlike. V m W be a relational morphism. Suppose Proof. Let ϕ : S → T ∈ V ψ : T → W ∈ W is a relational morphism with eψ −1 ∈ V for each idempotent e ∈ W . Then since U is W-idempotent pointlike, there is an idempotent e ∈ W such that U ≤ eψ −1 ϕ−1 . Let V = eψ −1 . Then V ∈ V and ϕ restricts to a relational morphism ρ : U → V . Since A is V-pointlike, we conclude m W-pointlike, there exist v ∈ V ≤ T with A ⊆ vρ−1 ⊆ vϕ−1 . Thus A is V as desired.
Corollary 2.3. Suppose S is a semigroup and G ≤ S is a group. Suppose m V-pointlike. that G is V-pointlike. Then G is A Proof. It was observed that G must in fact be V-idempotent pointlike. Since m V-pointlike. G is A-pointlike, Proposition 2.2 implies G is A We shall make use of the results of [19] on pointlikes for G ∗ V. We therefore refer the reader to that paper for the notions of a parameterized relational morphism Φ, the derived partial transformation semigroup DΦ of such a relational morphism and an admissible partition P on DΦ . The reader is also referred to [3] for the first two notions and to [14] for a summary of the principal results of [19]. We use the notation of [19]. The following lemma is the version from [19] of Rhodes’s “Tie Your Shoes” Lemma [2]. Lemma 2.4 (Tie Your Shoes). Suppose R is a regular R-class of a semigroup S belonging to a J -class J. Suppose J 0 = M0 (G, A, B, C) is a Rees coordinatization, Φ : (R, S) → (Q, T ) a parameterized relational morphism and P an admissible partition on DΦ . Let R be the R-class corresponding to a ∈ A. Suppose b1 Ca0 6= 0 6= b2 Ca0 and x = (a, g(b1 Ca0 )−1 , b1 ), y = (a, g(b2 Ca0 )−1 , b2 ) ∈ qϕ−1 1 . Then (x, q) P (y, q). We shall also need the following straightforward lemma. Lemma 2.5. Let Φ = (ϕ1 , ϕ2 ) : (X, S) → (Q, T ) be a parameterized relational morphism with T ∈ V. Let x ∈ X and suppose H ≤ S is Vidempotent pointlike. Then there is an idempotent e 0 ∈ E(T ) and q ∈ Q −1 0 such that H ≤ e0 ϕ−1 2 , qe = q and xH ⊆ qϕ1 .
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Proof. Since H is V-idempotent pointlike, there is an idempotent e0 of T 0 with H ≤ e0 ϕ−1 2 . Let q ∈ xϕ. Then 0 0 −1 xH ⊆ q 0 ϕ−1 1 H ⊆ (q e )ϕ1 .
Setting q = q 0 e0 completes the proof.
Now we turn to our main technical lemma that will allow us to calculate the complexity of Sn inductively. We use the notation established in the first part of Subsection 2.1, in particular G, H, F (G ∪ 0, g) and the matrix P. Lemma 2.6. Suppose G = hgi is a cyclic group and suppose F (G ∪ 0, g) is a subsemigroup of a semigroup S such that H ∈ PL V (S). Then G ∈ PLA m (G∗V) (S) (where we identify G with a subsemigroup of F (G ∪ 0, g) in our “canonical” way). Proof. Let R be the R-class of a0 × G × B in S. By Corollary 2.3, it suffices to show G is G ∗ V-pointlike. Since G = {1, g}n for n sufficiently large and G∗V-pointlikes are closed under products, it suffices to show Y = a0 ×{1, g}×0 is G∗V-pointlike. By the results of [19], it suffices to show that, for all parameterized relational morphisms Φ = (ϕ1 , ϕ2 ) : (R, S) → (Q, T ) with T ∈ V and for all admissible partitions P on DΦ , there exists q ∈ Q such that Y ⊆ qϕ−1 1 and Y × {q} is contained in a single block of P. So suppose Φ : (R, S) → (Q, T ) is a parameterized relational morphism with T ∈ V and let P be an admissible partition on DΦ . Since H is a group and is V-pointlike, our above observations show that H is in fact V-idempotent pointlike. Set x = (a0 , 1, 00 ), y = (a0 , 1, 20 ). Notice that xH = yH = {x, y}. So Lemma 2.5 shows that there exists q ∈ Q such that 0 0 x, y ∈ qϕ−1 1 . Since 0 P a6 = 1 = 2 P a6 , it follows from “Tie Your Shoes” that (x, q) P (y, q). Choose t0 ∈ tϕ2 . By Lemma 2.5, there exists e0 ∈ E(T ) with H ≤ e0 ϕ−1 2 . Set X = {x, y}tH. Since (2.5)
we see that
{x, y}t = {(a0 , g, 0), (a0 , 1, 2)},
X = a0 × G × {0, 1, 2, 3}, that = is defined and that X ⊆ q 0 ϕ−1 1 . Let e be the identity of H (and hence of F (G ∪ 0, g)). Consider q0
qt0 e0
(x, q)(q, (te, t0 e0 ), q 0 ) and (y, q)(q, (te, t0 e0 ), q 0 ).
Since P is an automaton congruence, (x, q) P (y, q) and te = t, it follows from (2.5) that ((a0 , g, 0), q 0 ) P ((a0 , 1, 2), q 0 ). Repeated application of “Tie Your Shoes” yields: ((a0 , g, 0), q 0 ) P ((a0 , g, 1), q 0 ) P ((a0 , g, 2), q 0 ) P ((a0 , g, 3), q 0 ) and ((a0 , 1, 2), q 0 ) P ((a0 , 1, 3), q 0 ) P ((a0 , 1, 0), q 0 ) P ((a0 , 1, 1), q 0 ).
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We conclude that (a0 × {1, g} × 0) × q 0 belongs to a single partition block of P, as desired. The reader is asked to review the notation of Subsection 2.1. Theorem 2.7. c(Sn ) = n + 1. Proof. By Proposition 2.1, c(Sn ) ≤ n+1. We prove by downwards induction on i that Gi is Cn−i -pointlike in Sn . Clearly Gn , being a group, is Apointlike in Sn . Assume Gi is Cn−i -pointlike in Sn . Then Gi is the group of units of Si ≤ Sn . Moreover F (Gi−1 ∪ 0, gi−1 ) is a subsemigroup of Si (and hence of Sn ). Lemma 2.6 with S = Sn , H = Gi , G = Gi−1 and g = gi−1 then allows us to conclude that Gi−1 ∈ PLA m (G∗Cn−i ) (Sn ). That is, Gi−1 Cn−(i−1) -pointlike in Sn . We now conclude that G0 is a Cn -pointlike subset of Sn and hence c(Sn ) > n, establishing the theorem. 2.3. The Type II subsemigroup of Sn . Our next goal is to prove that m G) m G. The following proposition was proved in [11]. Sn ∈ (A Proposition 2.8. There is a relational morphism ϕ : S1 → G ∈ G with m G and with G1 ϕ = 1. 1ϕ−1 ∈ A Proof. Let G = {1, −1} be a cyclic group of order 2. Define a relational morphism ϕ : S1 → G by x ∈ G1 1 xϕ = −1 x ∈ N1 G else.
Then, 1ϕ−1 = S1 \ N1 . Setting T = 1ϕ−1 , we see that T is a monoid consisting of a group of units G1 and a 0-minimal regular ideal with aperiodic idempotent-generated subsemigroup. It follows immediately from the results of [15] that KG (T ) ∈ A. Alternatively, one could directly apply Ash’s Theorem [1]. Thus ϕ is the desired relational morphism. m G) m G, all n ≥ 1. Theorem 2.9. Sn ∈ (A
Proof. We prove by induction on n that there is a relational morphism m G. The result will ϕ : Sn → G ∈ G such that Gn ϕ = 1 and 1ϕ−1 ∈ A then follow. The case n = 1 is Proposition 2.8. Suppose the result holds for m G. Let n. Let ψ : Sn → G ∈ G be such that Gn ψ = 1 and 1ψ −1 ∈ A G0 = {1, −1} be a cyclic group of order two. Define a relational morphism ϕ : Sn+1 → G0 × G by (1, 1) x ∈ Gn+1 (−1, 1) x ∈ N n+1 xϕ = 0 × sψ x = (a, s, b), s ∈ S G n 0 G × G x = 0.
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The only non-trivial verifications to show that ϕ is a relational morphism are of the form uϕxϕ ⊆ (ux)ϕ for u ∈ Nn or u = (a0 , s0 , b0 ) with s0 ∈ Sn and x = (a, s, b) with s ∈ Sn (or the dual situation). If ux = 0, things are trivial. If not, suppose first u ∈ Nn ; then the middle coordinate is either multiplied by 1 or by gn . Since Gn ψ = 1, in either case we have (ux)ϕ = G0 × sψ = uϕxϕ. If u = (a0 , s0 , b0 ) with s0 ∈ Sn and ux 6= 0, then ux = (a0 , s0 s, b) and so uϕxϕ = G0 × s0 ψsψ ⊆ G0 × (s0 s)ψ = (ux)ϕ,
as desired. Now Gn+1 ϕ = (1, 1), so to finish the proof it suffices to show T = (1, 1)ϕ−1 m G. Let K = 1ψ −1 ≤ Sn . By the induction hypothesis, belongs to A m G. Notice that K∈A Let
T = Gn+1 ∪ (M(K, A, B, C)/A × 0 × B).
U = Gn+1 ∪ M0 ({1}, A, B, C) ≤ T. Let K I be K with a new adjoined identity I. Consider the submonoid M of U × K I defined by M = (Gn+1 × I) ∪ (M0 ({1}, A, B, C) × K)
and the map α : M T given by
(g, I)α = g ((a, 1, b), k)α = (a, k, b) (0, k)α = 0. Then, since multiplication by Gn+1 on M(K, A, B, C) doesn’t change the middle coordinate and C consists of only zeroes and ones, α is an onto homomorphism. Hence T divides U × K I . m G, since K is. Thus to complete the proof that Clearly K I ∈ A m G it suffices to show that U ∈ A m G. Define β : U → Gn+1 by T ∈A ( x x ∈ Gn+1 xβ = Gn+1 else. Then 1β −1 = 1 ∪ M0 ({1}, A, B, C) ∈ A, establishing the Theorem.
Theorems 2.7 and 2.9 complete the proof of Theorem 1.1. References 1. C. J. Ash, Inevitable graphs: A proof of the Type II conjecture and some related decision procedures, Internat. J. Algebra and Comput. 1 (1991), 127–146. 2. B. Austin, K. Henckell, C. Nehaniv, and J. Rhodes, Subsemigroups and complexity via the presentation lemma, J. Pure Appl. Algebra 101 (1995), 245–289. 3. S. Eilenberg, Automata, Languages and Machines, Academic Press, New York, Vol A, 1974; Vol B, 1976. 4. K. Henckell, Idempotent pointlikes, Internat. J. Algebra Comput., to appear.
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5. K. Henckell, S. Margolis, J.-E. Pin, and J. Rhodes, Ash’s type II theorem, profinite topology and Mal’cev products: Part I, Internat. J. Algebra and Comput 1 (1991), 411–436. 6. J. Karnofsky and J. Rhodes, Decidability of complexity one-half for finite semigroups, Semigroup Forum 24 (1982), 55–66. 7. K. Krohn and J.Rhodes, Algebraic theory of machines, I: Prime decomposition theorem for finite semigroups and machines, Trans. Amer. Math. Soc. 116 (1965), 450–464. 8. K. Krohn and J. Rhodes, Complexity of finite semigroups, Annals of Math. 88 (1968), 128–160. 9. K. Krohn, J. Rhodes, and B. Tilson, Lectures on the algebraic theory of finite semigroups and finite-state machines, Chapters 1, 5-9 (Chapter 6 with M. A. Arbib) of The Algebraic Theory of Machines, Languages, and Semigroups, (M. A. Arbib, ed.), Academic Press, New York, 1968. 10. J. Rhodes, The fundamental lemma of complexity for arbitrary finite semigroups, Bull. Amer. Math. Soc. 74 (1968), 1104–1109. 11. J. Rhodes, Kernel Systems - A global study of homomorphisms on finite semigroups, J. Algebra 49 (1977), 1–45. 12. J. Rhodes, Infinite iteration of matrix semigroups. I. Structure theorem for torsion semigroups, J. Algebra 98 (1986), 422–451. 13. J. Rhodes, Infinite iteration of matrix semigroups. II. Structure theorem for arbitrary semigroups up to aperiodic morphism, J. Algebra 100 (1986), 25–137. 14. J. Rhodes and B. Steinberg, Krohn-Rhodes complexity pseudovarieties are not finitely based. Submitted. 15. J. Rhodes and B. Tilson, Improved lower bounds for the complexity of finite semigroups, J. Pure Appl. Algebra, 2 (1972), 13–71. 16. L. Ribes and P. A. Zalesski˘ı, On the profinite topology on a free group, Bull. London Math. Soc. 25 (1993), 37–43. 17. B. Steinberg, On pointlike sets and joins of pseudovarieties, Internat. J. Algebra Comput. 8 (1998), 203–231. 18. B. Steinberg, On an assertion of J. Rhodes and the finite basis and finite vertex rank problems for pseudovarieties, J. Pure Appl. Algebra 186 (2004), 91–107. 19. B. Steinberg, On aperiodic relational morphisms. Submitted. 20. B. Tilson, Depth decomposition theorem, Chapter XI in [3]. 21. B. Tilson, Complexity of semigroups and morphisms, Chapter XII in [3]. 22. B. Tilson, Categories as algebra: An essential ingredient in the theory of monoids, J. Pure and Appl. Algebra 48 (1987), 83–198. 23. Y. Zalcstein, Group-complexity and reversals of finite semigroups, Math. Systems Theory 8 (1974/75), 235–242. University of California at Berkeley, Berkeley, California 90720, USA and School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada E-mail address:
[email protected] and
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