Symmetry, Integrability and Geometry: Methods and Applications Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations

Ultradiscretization with negative values is a long-standing problem and several attempts have been made to solve it. Among others, we focus on the symmetrized max-plus algebra, with which we ultradiscretize the discrete sine-Gordon equation. Another ultradiscretization of the discrete sine-Gordon equation has already been proposed by previous studies, but the equation and the solutions obtained here are considered to directly correspond to the discrete counterpart. We also propose a noncommutative discrete analogue of the sine-Gordon equation, reveal its relations to other integrable systems including the noncommutative discrete KP equation, and construct multisoliton solutions by a repeated application of Darboux transformations. Moreover, we derive a noncommutative ultradiscrete analogue of the sine-Gordon equation and its 1-soliton and 2-soliton solutions, using the symmetrized max-plus algebra. As a result, we have a complete set of commutative and noncommutative versions of continuous, discrete, and ultradiscrete sine-Gordon equations.


Ultradiscretization and its problem
Ultradiscrete integrable systems are integrable systems where independent variables take values in Z, and dependent variables in the max-plus algebra R max = R ∪ {−∞}. Among them is the famous box-ball system [21], represented by the equation This may be understood as a transformation of addition into max operation and of multiplication into addition. Setting δ = e −1/ in (1.2) and applying ultradiscretization, we obtain (1.1). The problem is, however, that ultradiscretization cannot be applied to subtraction, which is of course necessary in many discrete integrable systems. The reason is as follows. If one wants to define an ultradiscrete version of subtraction, it is natural to solve the linear equation for x ∈ R max . This has no solution when a > b, and therefore subtraction cannot be defined in general.
Several attempts [8,11,15,18,19,23] have been made to solve this problem. We focus on the symmetrized max-plus algebra [1,2], denoted by uR in this paper. This algebra is an extension of R max and looks natural in the sense it traces the construction of Z from N 2 . Linear algebra over uR is also possible, and ultradiscretization with uR is presented in [4]. These theories of uR are mainly developed in the field of discrete event systems and seems little known to the field of integrable systems.

Contents of the paper
The discrete sine-Gordon equation [3,7] (1 − δ)τ m l τ m+1 l+1 = τ m l+1 τ m+1 l − δσ m l+1 σ m+1 l , (1 − δ)σ m l σ m+1 l+1 = σ m l+1 σ m+1 l − δτ m l+1 τ m+1 l has not been ultradiscretized until recent years because soliton solutions include subtraction or even complex numbers. The first attempt is made by Isojima et al. [9,10] where a τ -only trilinear equation is exploited to exclude subtraction. Here we propose another method to ultradiscretize the sine-Gordon equation which utilizes uR. The equation and the solutions are ultradiscretized keeping subtraction and complex numbers in a highly direct fashion. Noncommutative integrable systems have been drawing more interest in the last two decades. It is difficult to point out the first appearance of such systems, but the noncommutative KdV equation is already mentioned in [13]. The first discrete noncommutative integrable system is probably the noncommutative discrete KP equation [12,17]. Along this line, we propose a noncommutative discrete analogue of the sine-Gordon equation, explore relations to other integrable systems, and construct multisoliton solutions by the Darboux transformation. Moreover, we also propose a noncommutative ultradiscrete analogue of the sine-Gordon equation and explicitly derive 1-soliton and 2-soliton solutions by ultradiscretization with uR. As a result, we have a complete set of commutative and noncommutative versions of continuous, discrete, and ultradiscrete sine-Gordon equations.
The rest of the paper is organized as follows. In Section 2.1, the discrete sine-Gordon equation and 1-soliton and 2-soliton solutions are reviewed. Special solutions such as the travelingwave and kink-antikink solutions are presented probably for the first time. In Section 2.2, an ultradiscrete analogue of the sine-Gordon equation is proposed and the solutions are obtained. Because of ultradiscretization with uR, correspondence between the discrete and ultradiscrete systems are direct, which is also supported by figures.
In Section 3.1, a noncommutative discrete analogue of the sine-Gordon equation is proposed. A relation to other integrable systems including the noncommutative discrete KP equation is explained, and multisoliton solutions are constructed by a repeated application of Darboux transformations. In Section 3.2, a noncommutative ultradiscrete analogue of the sine-Gordon equation is proposed and 1-soliton and 2-soliton solutions are derived. Also figures of solutions for both equations are displayed.
In Section 4, concluding remarks and discussions are presented.

Discrete and ultradiscrete sine-Gordon equations
In this section, we first review the discrete sine-Gordon equation [3,7] and several results around it. Explicit calculation of the traveling-wave, kink-antikink, kink-kink, and breather solutions are probably presented for the first time.
Then, we propose an ultradiscrete analogue of the sine-Gordon equation. The solutions are obtained in two ways: by calculations completely inside uR, and by ultradiscretization with uR. The correspondence between the discrete and ultradiscrete systems is quite clear. Similarity of profiles of solutions is also visually confirmed by figures. Our formulation is different from Isojima et al. [9].

Discrete sine-Gordon equation
We review the three representations of the discrete sine-Gordon equation, their connection to the sine-Gordon equation, and some solutions for the sine-Gordon equation.
For any function f = f (l, m) over Z 2 , define shift operations by Inverse operations are denoted by Let τ = τ (l, m), σ = σ(l, m) be functions Z 2 → C. Date et al. [3] gave the discrete sine-Gordon equation (dsG) in the following form where δ ∈ C × is a parameter with a small absolute value. The vacuum solution is the simplest solution, other than the null solution τ = σ = 0. Calculating the cross product of the both sides of (2.1a), (2.1b), we have where w is defined by w = τ σ .

K. Kondo
If we introduce u defined by we have Each of (2.2) and (2.3) is also called the discrete sine-Gordon equation, where (2.3) is the original form discovered by Hirota [7].
This is known as a discrete analogue of the modified KdV equation [16], and its ultradiscretization is also known [20].
Assume u is also a function u(x, y) of continuum variables x, y ∈ R and has an expansion where u x = ∂u/∂x, etc. Connect l, m to x, y via the Miwa transformation u(x, y; l, m) = u(x + la, y + mb) where a, b ∈ R × are parameters. Then we have u lm − u l − u m + u = abu xy + (higher-order terms of a, b).
Setting δ = ab and taking the limit a, b → 0 of (2.3) successively, we obtain lim a,b→0 and thus the (continuous) sine-Gordon equation This is known to have following special types of solutions (see, for example, [6]): the travelingwave solution the kink-antikink solution u = 4 arctan sinh vy the kink-kink solution 6) and the breather solution where v, ω are constants.

1-soliton and 2-soliton solutions
Isojima et al. [9] have given the following conditions for τ and σ to be a 1-soliton or 2-soliton solution. As a 1-soliton solution, assume where c, p, q ∈ C × are constants. By substitution, the dispersion relation is found to be a necessary and sufficient condition. As a 2-soliton solution, assume where α ∈ C × is a constant. This time, the pair of the dispersion relation and the relation is a necessary and sufficient condition.   . These do not seem to be previously presented in the literature, including Hirota [7] and Isojima et al. [9]. Replacing c by ic in the 1-soliton solution, we obtain in the light-cone coordinates. This corresponds to the traveling-wave solution (2.4) for the sine-Gordon equation (if we restrict δ, c, p ∈ R × ).

Kink-antikink and kink-kink solutions
Set p 1 = q 2 in the 2-soliton solution. Then and thus p 2 = q 1 . Rewriting in the light-cone coordinates, we have We set . (2.14) This corresponds to the kink-antikink solution (2.5). Similarly, setting p 1 q 2 = 1 gives and thus p 2 q 1 = 1. We have for the same β, c 1 , c 2 defined above and .

Breather solution
Consider the kink-antikink solution where p 1 and p 2 are complex numbers satisfying Such p 1 , p 2 are complex conjugates of each other. If we write p 1 = g + hi, p 2 = g − hi and substitute these into (2.13), we find g, h must satisfy As a quadratic equation of g, the condition for the existence of real roots is given by Such a real number h does exist if δ ≥ 0, and g is given by Rewriting p 1 = re iθ , p 2 = re −iθ , we obtain β = iγ where γ is defined by

Ultradiscrete sine-Gordon equation
In order to ultradiscretize the discrete sine-Gordon equation (2.1a), (2.1b), we must deal with negative numbers since either or both of τ and σ include subtractions. We adopt ultradiscretization with the symmetrized max-plus algebra uR. For details, see Appendix A and references cited there. We perform ultradiscretization of dsG through the parametrization This can be regarded as an other aspect of continuum limit since δ → 0 as s → ∞. Assuming δ We call the pair (2.17a), (2.17b) the ultradiscrete sine-Gordon equation (udsG). The vacuum solution T = S = 0 is the simplest solution, other than the null solution T = S = −∞. We can also ultradiscretize (2.2) to obtain We also call (2.18) the ultradiscrete sine-Gordon equation. The ultradiscretization of (2.3) is unclear.

Deterministic time evolution and class of solutions
It seems sensible to restrict ourselves to the class of signed solutions, that is, T, S, W ∈ uC ∨ for any (l, m) ∈ Z 2 since it permits basic properties like weak substitution. The null and vacuum solutions are signed solutions.
The problem is that udsG no longer admits time evolution, at least deterministic one, in general, since the balance relation is not equality. For example, if we have we cannot determine f (t + 1) from f (t), since this relation is satisfied whenever |f (t + 1)| ⊕ ≤ 3. Strictly speaking, udsG is not an equation.
But in some cases, it actually becomes an equation, or furthermore, a deterministically evolutionary form. Multiplying T −1 to (2.17a) and S −1 to (2.17b), we have If T −1 T = S −1 S = 0 and the right hand sides are signed, we obtain by reduction of balances (see Appendix A). We call (2.19a), (2.19b) the deterministically evolutionary form of udsG. If we replace D by D and restrict ranges of D, T , S to R for example, the assumptions are satisfied, and we obtain the completely ordinary-looking ultradiscrete equation: Deterministic time evolution is also possible in other settings, which are presented in the following sections. It might be natural to think we should consider (2.19a), (2.19b), or even (2.20a), (2.20b) only. However, it seems that the former cannot capture the traveling-wave, kink-antikink, and kink-kink solutions. And the latter does not even seem to contain soliton solutions. Therefore, we consider (2.17a), (2.17b) primarily. We

1-soliton solution
Consider a signed solution T , S satisfying where C ∈ uC ∨ and P, Q ∈ uR ⊗ . Weakly substituting these into (2.17a), (2.17b), we have where 0 ⊕ D = 0 D = 0 is used. The dispersion relation is a sufficient condition for (2.22a), (2.22b) to hold, since we can construct them by adding and multiplying same numbers to the both sides of (2.23). Rewriting (2.23), we have and thus P = 0 or Q = 0. Obviously, (2.21) and (2.23) can be obtained by ultradiscretizing (2.8) and (2.9), respectively, through The solution is, however, not completely determined yet, because the balance relation is not equality as stated before. So we try to utilize reduction of balances. If C ∈ uZ is an odd number and P, Q ∈ uZ are even numbers, then F is always odd and 0 ⊕ F , 0 F can never be balanced since 0 is even. By reduction of balances, we obtain and W is also immediately determined since S −1 is signed. This solution admits deterministic time evolution since in the light-cone coordinates (2.11). It is somehow difficult to depict ultradiscrete numbers in figures; here signs and absolute values are displayed separately, and signs are mapped from 0, 0 • , ⊕0 to −1, 0, 1, respectively (balanced elements do not appear in the figure, though). Observe that the form of the 1-soliton solution w for dsG is preserved in the signs. Absolute values are always 0, corresponding to the fact that w asymptotically behaves as ±1.

Traveling-wave solution
If we replace C by CI and redefine F = CP l Q m (C ∈ uR ⊗ ), we obtain We choose odd C and even P , Q so that 0 F 2 is always signed and reduction of balances can be applied. This solution no longer admits deterministic time evolution, but is apparently ultradiscretization of the traveling-wave solution (2.12) for dsG. Fig. 2.8 shows the solution with The uRe and uIm parts are displayed separately. The profile of the traveling-wave solution for dsG is preserved well.

2-soliton solution
Assume where A ∈ uR ⊗ and F j = C j P l j Q m j . We also assume By substitution, we find the pair of the dispersion relation is a sufficient condition for (2.24) to become a solution. Obviously, (2.25) is ultradiscretization of (2.10). When P 1 = P 2 = 0 or Q 1 = Q 2 = 0, any A satisfies (2.25). When P 1 = Q 2 = 0, we have The case P 2 = Q 1 = 0 is similar. We can choose A, C j , P j , Q j ∈ uZ such that 0 ⊕ AF 1 F 2 is always positive, even and F 1 ⊕ F 2 is negative, odd. Then the solution is determined as This admits deterministic time evolution, of course. Fig. 2.9 shows the solution with

Kink-antikink and kink-kink solutions
If we replace C 1 by C 1 I, C 2 by C 2 I, and redefine We choose C j , P j , Q j ∈ uZ such that or are always signed and the balance relations become equalities.
If we set P 1 = Q 2 , P 2 = Q 1 , we have the kink-antikink solution. Similarly, setting gives the kink-kink solution. These solutions does not admit deterministic time evolution, but are ultradiscretization of (2.14), (2.15). The ultradiscretization of the breather solution is unclear. Fig. 2.10 shows the kink-antikink solution with and Fig. 2.11 shows the kink-kink solution with Observe that in the uIm part, two waves approach to each other for t < 0, collide at t = 0, and move away from each other for t > 0. In the kink-antikink solution, the two waves have the same sign and bump up by collision. In the kink-kink solution, the two have opposite signs and reflect by collision.

Noncommutative discrete and ultradiscrete sine-Gordon equations
In this section, we propose a noncommutative discrete analogue of the sine-Gordon equation as a compatibility condition of a certain linear system. This equation reduces to the commutative version once the underlying algebra turns out to be commutative and one simple reduction condition is applied. A reduction from the noncommutative discrete KP equation [12,17] also gives the equation, and continuum limit of the equation gives the noncommutative (continuous) sine-Gordon equation, which is already known in a different context [14]. We define the Darboux transformation, which constructs new solutions from old ones, and obtain Casoratian-type solutions by repeating it. Explicitly setting the starting solutions for repetition, we derive so-called multisoliton solutions. Along the construction of Casoratian-type solutions, quasideterminants [5] are used, which is a noncommutative extension of determinants. The theory needs some space for explanation, but it is not essential to the main story. Therefore, we only briefly explain the definition and some properties of them in Appendix B. For details, see [5].
We finally propose a noncommutative ultradiscrete analogue of the sine-Gordon equation. Noncommutative ultradiscrete setting is probably one of the hardest environments for integrable systems to exist, but we manage to obtain 1-soliton and 2-soliton solutions by ultradiscretization.
Notations are slightly changed in this section because of the complexity of expressions we are going to manipulate. Shifts are always indicated after a comma like f ,l . This is to distinguish indices and shifts. In addition, shift operators T l , T m are also used: Do not confuse these with the ultradiscretized τ function of the previous section; in the noncommutative setting, τ functions do not seem to exist. We also use superscripts for elements of matrices. For example, 3.1 Noncommutative discrete sine-Gordon equation

Linear system
Let w = w(l, m), v = v(l, m) be functions Z 2 → Mat(N, C) and where a, b, λ ∈ C × are parameters. Consider the linear system for φ, ψ : Z 2 → Mat(N, C). Denoting entrywise shift operations by T m B l = B l,m etc., we have These must coincide, so we require the compatibility condition This is equivalent to We call the pair (3.2a) and (3.2b) the noncommutative discrete sine-Gordon equation (ncdsG). gives the (commutative) discrete sine-Gordon equation [3,7] w ,lm
Then, setting we can rewrite (3.5) as By imposing the reduction condition w i (n 1 + 2, n 2 , n 3 ) = w i (n 1 , n 2 , n 3 ) and defining v i = w i,1 , we obtain Proof . Let us rewrite (3.6a) using only w 1 , v 1 . From (3.6e) we immediately have and thus try to rewrite the second term.

Continuum limit
Assume w is also a function w(x, t) of continuum variables x, t ∈ R and has an expansion where w x = ∂w/∂x, etc. Connect l, m to x, t via the Miwa transformation w(x, t; l, m) = w(x + la, t + mb).
Assume similarly for v = v(x, t; l, m). Then we have and from (3.2a), (3.2b) Taking the limit a, b → 0 successively, we obtain where u is defined by Proof . Under (3.13), (3.12b) clearly holds. And (3.14) is immediate from (3.12a) since
Proof . From the linear system (3.1), we can write Then we have 1 .

Multisoliton solutions
The simplest solution for (3.2a) and (3.2b) is the vacuum solution (w, v) = (1, 1). The linear system (3.1) of the vacuum solution is which has two basic solutions Let λ k (k = 1, 2, . . .) be mutually different eigenvalues and define  where c k ∈ Mat(N, C) are parameters introducing noncommutativity. t φ k ψ k is of course an eigenfunction of the vacuum solution for eigenvalue λ k . Repeating the Darboux transformation by t φ k ψ k , we can construct multisoliton solutions.
A 1-soliton solution is given by where f k is defined by As a concrete example, Fig. 3.1 shows the behavior of in the light-cone coordinates (2.11). A 2-soliton solution is given by

Casoratian-type solutions
Let (w, v) be a solution for (3.2a) and (3.2b), t φ k ψ k be eigenfunctions of (w, v) for eigenvalues λ k (k = 1, 2, . . .), where λ k are mutually different. Define repetition of the Darboux transformation by For notational convenience, we introduce reduced shift operator T defined by where f (x 1 , x 2 , . . .) is any rational function of noncommutative variables x j . For example, we have Proof . We prove by induction. Obviously, φ T n−1 ψ 1 T n−1 ψ 2 · · · T n−1 ψ n T n−1 ψ k T n ψ 1 T n ψ 2 · · · T n ψ n T n ψ k .

respectively.
Before proceeding to the proof, we prepare the following lemma.
Proof . By the column homological relation (Proposition B.3), we have By the assumption, we have C p 2 n = C p 2 n , C p 1 n = C p 1 n . Therefore, we obtain (3.18) by multiplying the inverse of (3.19a) to (3.19b) from the left.

K. Kondo
Proof of Theorem 3.7. We prove by induction. The case n = 0 is trivial.
k have the above expressions for certain n > 0. Then, By Lemma 3.8, we obtain We define an (n − 1) × (n − 1) matrix A 0 by With A 0 , the above quasideterminants are rewritten as

0
, and also, we have a trivial identity By the invariance under row and column permutations (Proposition B.2) and Sylvester's identity (Proposition B.4), we can combine these four quasideterminants into one to obtain With the same technique for w (n) , we obtain Similarly for ψ (n+1) k .

Ultradiscretization
We perform ultradiscretization of ncdsG by the parametrization We call the pair (3.20a) and (3.20b) the noncommutative ultradiscrete sine-Gordon equation (ncudsG). Because uMat(N, uC) can be realized by uMat (2N, uR), we use uMat(N, uR) as the underlying algebra for simplicity.

1-soliton solution
In order to ultradiscretize solutions for ncdsG, we introduce These solve the dispersion relation and any solution of (3.22) is parametrized by λ j through (3.21) unless ab = 1. As in the commutative case, (3.22) is ultradiscretized to where p j ud −→ P j , q j ud −→ Q j . We can directly discretize the 1-soliton solution (3.15a), (3.15b) to obtain This relation is valid, but inadequate to determine W , V in many cases. For simplicity, we assume N = 2 hereafter. If we write W = W ικ , F j = F ικ j , the (1, 2)-th element of (0 F 1 )(0 ⊕ F 1 ) −1 is given by and F 11 1 ⊕ exceeds 0 for large ±l or ±m. Then this element is balanced and W 12 cannot be determined. Therefore, we need more precise expressions to ultradiscretize. Define Of course, w = h 1 , v = g 1 is a 1-soliton solution for ncdsG. Writing f j = f ικ j , we have By ultradiscretization, we obtain We can choose C j , P j , Q j ∈ uZ such that 0 ⊕ F 11 j 0 F 22 j is always even and F 12 j F 21 j odd. Then all the elements on the r.h.s. of (3.24a), (3.24b) are signed and G j , H j are completely determined. Fig. 3.3 shows W = H 1 with

2-soliton solution
Ultradiscretization of (3.16a), (3.16b) gives In order to determine the value of L j , we examine the relation K. Kondo  By the dispersion relation (3.23), we have P j = 0 or Q j = 0. When Q j = 0, q j behaves like a constant with regard to the ultradiscretization parameter s and p j cannot behave like one. Therefore, we have Similarly, when P j = 0, we have If we choose P 2 = Q 1 = 0, we have |L 1 | ⊕ > |L 2 | ⊕ and thus  Similarly, if P 1 = Q 2 = 0, These are chosen so that every elements involved are signed.

Conclusion and discussion
We have proposed an ultradiscrete analogue of the sine-Gordon equation and constructed signed 1-soliton and 2-soliton solutions utilizing uR. The traveling-wave, kink-antikink, and kink-kink solutions, which contain ultradiscrete complex numbers, do exist and their correspondence to those for the discrete sine-Gordon equation is quite clear. When the range of solutions are restricted to uR, even deterministic time evolution is possible. As stated in Section 1, another ultradiscretization of the sine-Gordon equation has been given by Isojima et al. [9,10]. There, only τ is ultradiscretized, no complex solutions are dealt with, and time evolution is not possible. Our formulation looks better in these respects. Also, ultradiscretization via the parametrization (2.16) can be considered as another aspect of continuum limit, but ultradiscretization in [9] cannot since they choose a parametrization such that δ → ±1.
We have also proposed a noncommutative discrete analogue of the sine-Gordon equation and revealed its relation to other integrable systems including the noncommutative discrete KP equation. Also, multisoliton solutions are constructed by a repeated application of Darboux transformations. And finally, a noncommutative ultradiscrete analogue of the sine-Gordon equation and its signed 1-soliton and 2-soliton solutions are derived by ultradiscretization with uR.

A Symmetrized max-plus algebra and ultradiscretization
We make extensive use of the symmetrized max-plus algebra uR in the main part of the paper. Therefore we describe basic definitions and properties of uR here. For details, see Baccelli et al. [2].
We write x y for x ⊕ ( y), which is regarded as subtraction. Define absolute value | | ⊕ : R 2 max → R max by Define balance operator • by

A.1.2 Symmetrized max-plus algebra
It is natural to consider the balance relation ∇ defined by ∇ is reflexive and symmetric, but not transitive. Therefore, we introduce another relation R defined by y 2 ), when x 1 = x 2 and y 1 = y 2 , (x 1 , x 2 ) = (y 1 , y 2 ), otherwise.
R is an equivalence relation compatible with the operations ⊕, ⊗, , | | ⊕ , • , and the relation ∇. Thus, we can define the quotient structure uR = R 2 max / R . This is called the symmetrized max-plus algebra [1,2]. Usually this is denoted by S, but we use uR to imply it is somehow a whole set of ultradiscrete real numbers. We will also introduce uZ, uC later.
Proposition A.1. We have three kinds of equivalence classes: and (−∞, −∞) is the only element which belongs to any two of the three sets. Thus, we simply write x for (x, −∞), x for (−∞, x), and x • for (x, x). Define sign function sgn x by x ∈ uR is said to be positive if sgn x = 0, negative if sgn x = 0, and balanced if sgn Proposition A.2. Let uR ⊗ denote the whole set of invertible elements in uR. Then, with obvious notations. uZ is a subdioid of uR and can be regarded as a whole set of ultradiscrete integers. x ∈ uZ is said to be even if |x| ⊕ is even, odd if |x| ⊕ is odd. We do not define whether −∞ is even or odd. We have of course

A.1.3 Properties of balance relation
We make much use of ∇, rather than R, since members of R • max can be regarded as a kind of null elements by virtue of the following proposition.
Proposition A.4. For any x ∈ uR and t ∈ R max , Proposition A.5. For any x, y ∈ uR, we have Proposition A.6. For any x, y, z, w ∈ uR, we have Proposition A.7 (weak substitution).

A.1.4 Matrices and determinants
Let uMat(N, uR) denote the whole set of N × N matrices over uR. Define addition ⊕ by and multiplication ⊗ by for any (a ij ), (b ij ) ∈ uMat(N, uR). Then uMat(N, uR) becomes a dioid, noncommutative when N > 1. , • , and ∇ are of course defined by respectively. (a ij ) ∈ uMat(N, uR) is said to be signed if all the elements are signed. The whole set of signed elements in uMat(N, uR) is denoted by uMat(N, uR) ∨ . uR is embedded into uMat(N, uR) by For any permutation σ ∈ S N , define sgn(σ) by sgn(σ) = 0, when σ is even, 0, when σ is odd.
And define the determinant of a matrix A = (a ij ) ∈ uMat(N, uR) by det A is also denoted by |A| or |a ij |.
Let cof ij (A) denote the cofactor of a ij in |A|, which by definition satisfies for any j. Define the adjacent matrix of A by Theorem A.14.
If |A| ∈ uR ⊗ , define A −1 by This is not a multiplicative inverse in general, but plays a similar role with regard to ∇. Therefore we use the notation A −1 .

A.1.5 Ultradiscrete complex numbers
It is well known that we can construct complex numbers by 2 × 2 real matrices, using as the imaginary unit. Here we try to construct ultradiscrete complex numbers in a similar way. Let uMat(N, uR) denote the algebra of N × N matrices whose elements are in uR. Define I ∈ uMat(2, uR) by We have Define uC ⊂ uMat(2, uR) by Proposition A.15. uC is a commutative subdioid of uMat(2, uR).
And uC is commutative because I 0 and I 1 are commutative.
When z ∈ uC is expressed as z = x + yI where x, y ∈ uR, we write uRe z = x, uIm = y.
The whole set of signed elements of uC is denoted by uC ∨ . If det(x ⊕ yI) = x 2 ⊕ y 2 ∈ uR ⊗ , we have

A.2 Ultradiscretization with negative numbers
Ultradiscretization with negative numbers is presented in De Schutter et al. [4]. Here we reformulate it in a similar, but more convenient form for our purpose. Let f (s) and g(s) be real functions. We say f (s) is asymptotically equivalent to g(s) if there exists a real number s 0 such that g(s) = 0 for any s > s 0 and lim s→∞ f (s) g(s) = 1.
We also say f (s) is asymptotically equivalent to 0 if there exists a real number s 1 such that f (s) = 0 for any s > s 1 . Asymptotic equivalence is an equivalence relation and denoted by f (s) ∼ g(s).
We regard 0 ∼ µe (−∞)s for some µ ∈ R × and 0 ud −→ −∞ as a convention. It is very important here to notice that µ F is not restricted to positive numbers, unlike the usual ultradiscretization procedure.

A.3 Ultradiscretization of matrices and complex numbers
We also reformulate ultradiscretization of matrices in [4]. Extension to complex numbers is straightforward.

B Quasideterminants
Quasideterminants [5] are noncommutative extension of determinants, or, more precisely, determinants divided by cofactors. Here we describe the definition and some properties required for Theorem 3.7. See [5] for more detail. Let R be a ring and Mat(N, R) be the whole set of N × N matrices over R. R is not commutative in general. For any (a ij ), (b ij ) ∈ Mat(N, R), define addition by (a ij ) + (b ij ) = (a ij + b ij ) and multiplication by Ordering of multiplication is important here. For any A = (a ij ) ∈ Mat(N, R), define the (p, q)-th quasideterminant |A| pq by |A| pq = a pq − r q p (A pq ) −1 c p q , where r q p is the p-th row of A without the q-th element, c p q is the q-th column of A without the p-th element, and A pq is A without the p-th row and the q-th column. |A| pq is also written as |A| pq = a 11 · · · a 1N . . .