IDENTITIES IN UPPER TRIANGULAR TROPICAL MATRIX SEMIGROUPS AND THE BICYCLIC MONOID
arXiv:1612.04219v1 [math.RA] 13 Dec 2016
LAURE DAVIAUD1, MARIANNE JOHNSON2 and MARK KAMBITES3 Abstract. We answer a question of Izhakian and Margolis, by showing that the identities which hold in the monoid of 2 × 2 upper triangular tropical matrices are exactly the same as those which hold in the bicyclic monoid.
1. Introduction Over the last few years there has been considerable interest in the structure of the monoid Mn (T) of tropical (max-plus) matrices under multiplication, with lines of investigation including characterisations of Green’s relations [7, 8, 15, 16], structural properties of subsemigroups and maximal subgroups [5, 6, 13], as well as connections between the algebraic properties of elements (or indeed subsemigroups) and their actions upon tropically convex sets [12, 18, 26]. These matrix semigroups are also closely related to max-plus automata [3]. A natural question, especially in view of d’Alessandro and Pasku’s proof that finitely generated subsemigroups of Mn (T) have polynomial growth [5], is whether Mn (T) satisfies a semigroup identity. Izhakian and Margolis [14] were the first to consider this question, in the 2 × 2 case; they showed that M2 (T) does indeed satisfy an identity. A key step of their proof was establishing an identity for the upper triangular submonoid U T2 (T), which is also of interest in its own right. Specifically, they showed that U T2 (T) satisfies the celebrated identity AB 2 A2 BAB 2 A = AB 2 ABA2 B 2 A, which was shown by Adjan [1] to hold in the bicyclic monoid B. Since B embeds in U T2 (T), every identity satisfied in the latter must also hold in the former. In view of this and their results, Izhakian and Margolis posed the natural question of whether the converse holds, that is, whether U T2 (T) and B satisfy exactly the same identities, or equivalently (by Birkhoff’s theorem [2]), whether they generate the same variety. Further evidence that this might be the case was recently provided by Chen, Hu, Luo and Sapir [4], who showed that the variety generated by U T2 (T) shares with that generated by B the property of not admitting a finite basis of identities. 1
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland. Email
[email protected]. Work supported by the LIPA project, funded by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 683080). 2 School of Mathematics, University of Manchester, Manchester M13 9PL, UK. Email
[email protected]. 3 School of Mathematics, University of Manchester, Manchester M13 9PL, England. Email
[email protected]. 1
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IDENTITIES IN TROPICAL MATRIX SEMIGROUPS
In Section 3 we develop an exact characterisation of identities which hold in U T2 (T), by associating k tropical polynomial equations in k variables to each identity on k letters. This gives a simple (and algorithmically efficient) method to check whether any given identity holds. In Section 4, by considering the embedding of B into U T2 (T), we use this to show that an identity which fails to hold in U T2 (T) must also fail in B, thus answering the abovementioned question of Izhakian and Margolis [14] and establishing our main theorem: Main Theorem. The bicyclic monoid B and the upper triangular tropical matrix monoid U T2 (T) satisfy exactly the same semigroup identities. It follows by Birkhoff’s Theorem, of course, that B and U T2 (T) generate the same variety, and hence share all properties of monoids which are visible in the variety generated. For example, by a theorem of Shneerson [24], B has infinite axiomatic rank, so U T2 (T) must also have. (In particular, since infinite axiomatic rank implies the non-existence of a finite basis, we can deduce the results of [4] from our theorem plus [24].) We mention also work of Pastijn [20], which gives an efficient algorithm for checking whether identities hold in B by reducing to checking linear inequalities or, equivalently, properties of certain polyhedral complexes. Once we have established that identities in B are equivalent to identities in U T2 (T), it becomes evident that our methods (for identities in U T2 (T)) are related to Pastijn’s (for identities in B). However, since our methods are used to establish this theorem in the first place, we cannot hope to deduce our results from Pastijn’s work. In higher dimensions, understanding identities seems to be hard. Izhakian and Margolis have conjectured that Mn (T) satisfies an identity for every n; evidence supporting this includes their own results on M2 (T), and the proof by d’Alessandro and Pasku that finitely generated subsemigroups of Mn (T) all have polynomial growth. Shitov [22] has recently produced an identity for M3 (T), over a 2-letter alphabet and with 1795308 letters on each side, while Izhakian [11] has shown that matrices of maximal rank satisfy certain identities. Again, the proof uses a reduction to the upper triangular case. Izhakian [9] claimed a family of identities for U Tn (T) for each n; in fact the proof contains a technical error and at least some of the claimed identities do not hold, but they can be corrected [10, 27] to give examples of identities which do hold in these semigroups. Further examples of identities for the upper triangular case have been constructed by Okninski [19]. A full understanding of all identities in Mn (T), or even U Tn (T), for n ≥ 3 seems still quite distant. 2. Tropical matrices semigroups and identities The tropical semiring T is the commutative idempotent semiring whose elements are drawn from the set R ∪ {−∞}, and whose binary operations are defined by a ⊕ b = max(a, b) and a ⊗ b = a + b, where −∞ should be thought of as the “zero” element of the semiring, satisfying −∞ ⊕ a = a and −∞ ⊗ a = −∞ for all a ∈ T. Often it will be convenient to work without this zero element; in order to distinguish this algebraic structure
IDENTITIES IN TROPICAL MATRIX SEMIGROUPS
3
from the field R we shall therefore write FT to denote the set R, under the operations ⊕ and ⊗. It is easy to see that the set of all n × n matrices with entries in T (respectively FT) forms a monoid (respectively semigroup) under the matrix multiplication induced from these operations. We denote these semigroups by Mn (T) and Mn (FT). We also write U Tn (T) (respectively U Tn (FT)) to denote the subsemigroup of Mn (T) consisting of those uppertriangular matrices in Mn (T) whose entries on and above the main diagonal lie in T (respectively FT). We write N and N0 respectively for the natural numbers with and without 0. If Σ is a finite alphabet, then Σ+ will denote the free semigroup on Σ, that is, the set of finite, non-empty words over Σ under the operation of concatenation. For w ∈ Σ+ and s ∈ Σ we write |w| for the length of w and |w|s for the number of occurrences of the letter s in w. The content of w is the map Σ → N0 , s 7→ |w|s . Recall that a (semigroup) identity is a pair of words, usually written “u = v”, in the free semigroup Σ+ on an alphabet Σ. We say that the identity holds in a semigroup S (or that S satisfies the identity) if every morphism from Σ+ to S maps u and v to the same element of S. If a morphism maps u and v to the same element we say that it satisfies the given identity in S; otherwise it falsifies it. (Monoid identities are defined similarly using the free monoid Σ∗ in place of Σ+ ; we shall work primarily with semigroup identities, but it is easy to deduce corresponding results about monoid identities from these.) For some purposes it is easier to work with Mn (T) or U Tn (T), while for others it is easier to work with Mn (FT) or U Tn (FT). The following proposition ensures that we can choose to work with whichever is easier at each stage. Proposition 2.1. For all n ∈ N, • Mn (T) and Mn (FT) satisfy exactly the same semigroup identities; and • U Tn (T) and U Tn (FT) satisfy exactly the same semigroup identities. Proof. Since Mn (FT) embeds in Mn (T), any identity satisfied in the latter is also satisfied in the former. Conversely, suppose an identity u = v over alphabet Σ is not satisfied in Mn (T). Let φ : Σ+ → Mn (T) be a morphism falsifying the identity, and choose i, j with φ(u)i,j 6= φ(v)i,j . Define a map from ψ from Σ+ to the semigroup Mn (T[x]) of matrices of formal polynomials in one variable x over T, by for each a ∈ Σ setting ψ(a) to be the matrix obtained from φ(a) by replacing each −∞ entry with x. For any x ∈ T, because evaluation at x is a semiring morphism from T[x] to T, we may define a semigroup morphism ψx : Σ+ → Mn (T) by ψx (w)i,j = [ψ(w)i,j ](x). If x is finite then the image of ψx is contained in Mn (FT), while if x = −∞ we recover the original morphism φ. Consider now the polynomials ψ(u)i,j and ψ(v)i,j . Since at x = −∞ they evaluate to φ(u)i,j and φ(v)i,j respectively, which are different, they must have different constant terms. It follows that for x small enough but finite, they evaluate to different values. Thus, for small enough x, ψx is a morphism to Mn (FT) falsifying the identity u = v.
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IDENTITIES IN TROPICAL MATRIX SEMIGROUPS
The proof in the upper triangular case is almost identical, except that in the construction of ψ only those −∞ entries on or above the main diagonal are replaced with the variable x. The following two elementary results will allow us at various points to simplify the structure and number of parameters in the matrices we must consider. Lemma 2.2. Let φ : Σ+ → Mn (T) a morphism, µs ∈ FT for each s ∈ Σ, and define a new morphism ψ : Σ+ → Mn (T) by ψ(s) = µs ⊗ φ(s) for all s ∈ Σ. If u and v are words with the same content and φ(u) = φ(v), then ψ(u) = ψ(v). Proof. Since tropical scaling commutes with matrix multiplication it is easy to see that O O s µ|v| ψ(u) = µs|u|s φ(u) = s φ(v) = ψ(v) s∈Σ
s∈Σ
where the powers are to be interpreted tropically, and the middle equality holds because φ(u) = φ(v) and u and v have the same content. Lemma 2.3. Suppose an identity u = v over alphabet Σ is satisfied by all morphisms φ : Σ+ → U Tn (FT) with the property that φ(s)n,n = 0 for all s ∈ Σ. If u and v have the same content then the identity u = v holds in U Tn (T). Proof. Every morphism to U Tn (FT) can clearly be obtained from one of the given form by the construction in Lemma 2.2, which since u and v have the same content means it satisfies the identity. So U Tn (FT) and hence also (by Proposition 2.1) U Tn (T) satisfy the identity. 3. Identities in U T2 (T). In this section we shall give an elementary but powerful characterisation of the identities which hold in U T2 (T) (or equivalently, by Proposition 2.1, in U T2 (FT)). Let w be a word over an alphabet Σ. For s ∈ Σ and 0 ≤ i ≤ |w| we write s λi (w) for the number of occurrences of the letter s within the first i letters of the word w (so in particular λs0 (w) = 0). For each t ∈ Σ we define a tropical polynomial function M O λs (w) xs i−1 , ftw : RΣ → T, x 7→ wi =t s∈Σ
where of course the powers are to be interpreted tropically, and a maximum over the empty set is taken to be −∞. We shall see shortly (Theorem 3.3) that the values of the polynomials ftw exactly characterise the identities which hold in U T2 (T). The following lemma explains how they arise from the multiplication in U T2 (FT): Lemma 3.1. Suppose φ : Σ+ → U T2 (T) is such that φ(s)1,1 ∈ FT and φ(s)2,2 = 0 for all s ∈ Σ, say xs x′s φ(s) = , where xs , x′s ∈ T with xs 6= −∞. −∞ 0
IDENTITIES IN TROPICAL MATRIX SEMIGROUPS
5
Then for any word w ∈ Σ+ , ϕ(w)1,2 =
M
x′s ⊗ fsw (x)
s∈Σ
Proof. This is a straightforward calculation, using the definitions of matrix multiplication and of the polynomial functions fsw . The next lemma says that the polynomial functions ftw are together sufficient to characterise the content of the word w. Lemma 3.2. If ftw (x) = ftv (x) for all t ∈ Σ and x ∈ RΣ then w and v have the same content. Proof. It is easy to see that ftw is the constant −∞ function if and only if w does not contain any occurrences of the letter t. Thus, we may assume that exactly the same letters from Σ occur in w and v. Let z ∈ Σ, and suppose that z occurs in both w and v. Define x ∈ RΣ by xz = 1 and xs = 0 for s ∈ Σ \ {z}. Then M O λs (w) M xs i−1 = λzi−1 (w) fzw (x) = wi =z s∈Σ
wi =z
where the last equality is because of the values of xs . Since for fixed z and w the function λzi (w) is non-decreasing with i, it is clear that the maximum on the right is attained when i is the position of the final z in w. With this value of i, we see that fzw (x) = λzi−1 (w) = |w|z − 1. By the same argument, fzv (x) = |v|z − 1. But by assumption fzw (x) = fzv (x), so we must have |w|z = |v|z . We are now ready to prove the main theorem of this section. Theorem 3.3. The identity w = v over alphabet Σ is satisfied in U T2 (T) if and only if ftw (x) = ftv (x) for all t ∈ Σ and x ∈ RΣ . Proof. Suppose first that we can choose x ∈ RΣ and t ∈ Σ, such that ftw (x) 6= ftv (x). Define a morphism xa x′a + ϕ : Σ → U T2 (T), a 7→ −∞ 0 where x′t = 0 and x′s = −∞ for each s 6= t. Then by Lemma 3.1, M ϕ(w)1,2 = x′s ⊗ fsw (x) = ftw (x) s∈Σ
and similarly ϕ(v)1,2 =
M
x′s ⊗ fsv (x) = ftv (x) 6= ϕ(w)1,2
s∈Σ
so the morphism ϕ falsifies the identity in U T2 (T). Conversely, suppose that ftw (x) = ftv (x) for all t ∈ Σ and all x ∈ RΣ . By Lemma 3.2, w and v have the same content. Hence, by Lemma 2.3 it suffices to show that the identity w = v is satisfied by morphisms ϕ : Σ+ → U T2 (FT) such that ϕ(s)2,2 = 0 for all s ∈ S. Let ϕ be such a morphism, and for each s ∈ Σ write xs = ϕ(s)1,1 and x′s = ϕ(s)1,2 .
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IDENTITIES IN TROPICAL MATRIX SEMIGROUPS
It is immediate from the properties of ϕ that ϕ(w)2,2 = 0 = ϕ(v)2,2 . It is also easy to see that for any word u, ϕ(u)1,1 =
|u| O
ϕ(ui )1,1 =
i=1
X
|u|s ϕ(s)1,1 .
s∈Σ
Since w and v have the same content, it follows that ϕ(w)1,1 = ϕ(v)1,1 . Finally, Lemma 3.1 gives M M ϕ(w)1,2 = x′t ⊗ ftu (x) = x′t ⊗ ftv (x) = ϕ(v)1,2 t∈Σ
t∈Σ
so that ϕ(w) = ϕ(u) as required.
4. The Bicyclic Monoid The bicyclic monoid B = hp, q | pq = ei is an inverse monoid which is ubiquitous in almost all areas of infinite semigroup theory, and indeed in many other areas of mathematics. Identities in the bicyclic monoid have been extensively studied; Adjan [1] established that the identity AB 2 A2 BAB 2 A = AB 2 ABA2 B 2 A is a minimal length identity for this monoid. Shleifer [23] has shown (with computer assistance) that there is just one other identity of minimal length. It follows from results of Scheiblich [21] that the free monogenic inverse monoid also satisfies the same identities as the bicyclic monoid. Jones [17] has shown that a semigroup variety contains a simple semigroup that is not bisimple if and only if it contains the bicyclic smeigroup. Motivated by this work, Pastijn [20] gave an elegant geometric characterisation of the identities which hold in terms of the properties of associated polyhedral complexes. Most recently Shneerson [25] has provided a family of interesting examples of semigroups of cubic growth which satisfy the same identities as the bicyclic monoid. In this section we apply Theorem 3.3 to prove that the identities satisfied by U T2 (T) are precisely those satisfied by the bicyclic monoid, thus establishing that U T2 (T) and several other associated semigroups all generate the same variety as B and the free monogenic inverse semigroup. Main Theorem. The bicyclic monoid B and the upper triangular tropical matrix monoid U T2 (T) satisfy exactly the same semigroup identities. Proof. It is well known and easy to see that every element of the bicyclic monoid can be written uniquely in the form q i pj for some i, j ∈ N0 . Izhakian and Margolis [14] noted that the map: i−j i+j i j ρ : B → U T2 (T), q p 7→ −∞ j − i is a monoid embedding of B into U T2 (T); of course, it also gives an embedding into U T2 (FT). It is immediate from the existence of this embedding that any identity which holds in U T2 (T) must hold in B. It remains to show that any identity which holds on B must also hold in U T2 (T). We shall show the contrapositive. To this end, suppose that a semigroup or monoid identity w = v over alphabet Σ does not hold in U T2 (T). Clearly we may assume that w and
IDENTITIES IN TROPICAL MATRIX SEMIGROUPS
7
v have the same content; indeed if z ∈ Σ is such that |w|z 6= |v|z then the identity is falsified in B by (for example) the morphism sending z to p and every other letter to the identity element. By Theorem 3.3 there exists x ∈ RΣ and t ∈ Σ such that ftw (x) 6= v ft (x). Suppose without loss of generality that ftw (x) > ftv (x). Since both functions are piecewise linear in the components of x ¯, there is an open Σ neighbourhood (in the usual Euclidean topology on R considered as a set of finite dimensional vectors) of x in which every vector satisfies the given inequality. This neighbourhood must contain a vector in QΣ so by replacing x with such a vector, we may assume without loss of generality that the components of x are all rational. Let d denote the least common multiple of the denominators in the lowest forms of the entries xs for s ∈ Σ. It is clear from basic properties of tropical polynomials that ftw (2dx) > fzw (2dx). Hence, by replacing x with 2dx, we may further assume, again without loss of generality, that the components of x are even integers. Now for each s ∈ Σ, choose a non-negative even integer x′s greater than xs , in such a way that x′t is very large relative to the other x′s s. Define a morphism ψ : Σ+ → U T2 (FT) by xa x′a ψ(a) = −∞ 0 By Lemma 3.1 we have ψ(w)1,2 =
M
x′s ⊗ fsw (x) = x′t ⊗ ftw (x)
s∈Σ
provided
x′t
was chosen sufficiently large, and similarly M ψ(v)1,2 = x′s ⊗ fsv (x) = x′t ⊗ ftv (x) 6= ϕ(w)1,2 s∈Σ
where the inequality is because ftw (x) 6= ftv (x). So the morphism ψ falsifies the identity in U T2 (FT). Next let is = 21 x′s , and js = is − 12 xs . Then is is non-negative (because x′s is non-negative) and integer (because x′s is even), while js is non-negative (because x′s > xs ) and integer (because is is integer and xs is even). Define a morphism φ : Σ+ → im ρ ⊆ U T2 (T) by is − js is + js xs x′s is j s φ(s) = ρ(q p ) = = (js − is ) ⊗ −∞ js − is −∞ 0 where the last equality follows from the definitions of is and js . Because of the last equality, we see that ψ can be obtained from φ by the construction in Lemma 2.2, with µs = is − js for each s. Hence, by the contrapositive of Lemma 2.2, φ falsifies the identity. But im φ ⊆ im ρ which is isomorphic to B, so the identity cannot hold in B. Corollary 4.1. The submonoids of U T2 (T) obtained by restricting the onand-above diagonal entries to be respectively in Q, Z, N0 , Q ∪ {−∞}, Z ∪ {−∞} and N0 ∪ {−∞} all satisfy exactly the same identities as B and U T2 (T).
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IDENTITIES IN TROPICAL MATRIX SEMIGROUPS
Proof. Write U T2 (X) for the given monoids, where X = Q, Z, N0 , Q∪{−∞}, Z ∪ {−∞} or N0 ∪ {−∞}. The monoids U T2 (Q), U T2 (Z), U T2 (Q ∪ {−∞}) and U T2 (Z ∪ {−∞}) are all contained in U T2 (T) and contain the image of the embedding ρ, which is a isomorphic to B. The monoids U T2 (N0 ) and U T2 (N0 ∪{−∞}) are again contained in U T2 (T), and so satisfy all of the given identities. Conversely, notice that every matrix in U T2 (Z) is a tropical scaling of a matrix in U T2 (N0 ). By the previous paragraph, any identity which does not hold in U T2 (T) is falsified by some morphism to U T2 (Z), and now the contrapositive of Lemma 2.2 yields a morphism falsifying the identity with image contained in U T2 (N0 ), and hence also in U T2 (N0 ∪ {−∞}). References [1] S. I. Adjan, Defining relations and algorithmic problems for groups and semigroups. Proceedings of the Steklov Institute of Mathematics, No. 85 (1966). (Translated from the Russian by M. Greendlinger American Mathematical Society, Providence, R.I. 1966.) [2] G. Birkhoff, On the structure of abstract algebras. Proc. Camb. Phil. Soc. 31 (1935), 433–454. [3] T. Colcombet, L. Daviaud, F. Zuleger, Size-change abstraction and max-plus automata. Mathematical foundations of computer science Part I, 208–219, Lecture Notes in Comput. Sci., 8634, Springer, Heidelberg, 2014. [4] Y. Chen, X. Hu, Y. Luo, O. Sapir, The finite basis problem for the monoid of twoby-two upper triangular tropical matrices. Bull. Aust. Math. Soc. 94 (2016), no. 1, 54–64. [5] F. d’Alessandro, E. Pasku, A combinatorial property for semigroups of matrices. Semigroup Forum 67 (2003), no. 1, 22–30. [6] S. Gaubert, On the Burnside problem for semigroups of matrices in the (max,+) algebra. Semigroup Forum 52 (1996), no. 3, 271-292. [7] S. Gaubert, Two lectures on max-plus algebra. Proceedings of the 26th Spring School of Theoretical Computer Science Algbres Max-Plus et applications en informatique et automatique (1998). [8] C. Hollings, M. Kambites, Tropical matrix duality and Green’s D relation. J. London Math. Soc. 86, no. 2 (2012) 520–538. [9] Z. Izhakian, Semigroup identities in the monoid of triangular tropical matrices. Semigroup Forum 88 (2014), no. 1, 145–161. [10] Z. Izhakian, Erratum to: Semigroup identities in the monoid of triangular tropical matrices Semigroup Forum 92 (2016), no. 3, p733. [11] Z. Izhakian, Semigroup identities of tropical matrix semigroups of maximal rank. Semigroup Forum 92 (2016), no. 3, 712–732. [12] Z. Izhakian, M. Johnson, M. Kambites, Pure dimension and projectivity of tropical polytopes. Advances in Mathematics 303 (2016), 1236–1263. [13] Z. Izhakian, M. Johnson, M. Kambites, Tropical matrix groups. Semigroup Forum, to appear. [14] Z. Izhakian, S. Margolis, Semigroup identities in the monoid of two-by-two tropical matrices. Semigroup Forum 80 (2010), no. 2, 191–218. [15] M. Johnson, M. Kambites, Multiplicative structure of 2x2 tropical matrices. Linear Algebra and its Applications, 435 (2011), 1612–1625. [16] M. Johnson, M. Kambites, Green’s J -order and the rank of tropical matrices. Journal of Pure and Applied Algebra, 217 (2013), 280–292. [17] P. R. Jones, Analogues of the bicyclic semigroup in simple semigroups without idempotents. Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), no. 1-2, 11–24. [18] G. Merlet, Semigroup of matrices acting on the max-plus projective space. Linear Algebra Appl. 432 (2010), no. 8, 1923–1935.
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