Tropical algebra with high-order matrix for multiple-noise removal

Abstract

The technology for multiple-noise removal has triggered skyrocketing interest in both mathematics and engineering, and the tropical algebra has laid the foundation for an abundance of noise filters. However, the denoising of the filter based on the traditional algebra is inextricably complex, and its algorithm is extremely intricate and awfully inefficient, so it is necessary to estimate the statistical characteristics of noise in a novel way. Now the tropical algebra has opened the path for a new way to design optimally a denoising method, which has obvious advantages over traditional ones in denoising efficiency and simple filtering algorithm. In this paper, the idempotent of the multiplicative semigroup of $4\times 4$ tropical matrices is studied. First, the tropical algebra and the tropical matrix multiplicative semigroup are introduced. Second, the idempotent classification of the $4\times 4$ tropical matrix multiplicative semigroup is given. Finally, an example is given, in which the filter based on tropical algebra is used for denoising, and an optimization method with tropical polynomials as constraints is given.

Introduction

Multiple noises appear everywhere in engineering, ¹ and the multiple-noise removal technology has been caught much attention, ² especially in acoustic recognition, ³ medical diagnosis,^4–6 image processing,^7–10 medical imaging,^11,12 and machine learning. ¹³ Multiple-noise removal is also required in nuclear magnetic resonance ¹⁴ and synthetic aperture radar. ¹⁵ In a multiple-noise model, a recorded image g is the multiplication of an original image u and a noise v:

$g=uv$

(1)

There are many practical applications involving multiple-noise models. ¹⁶ For instance, when monochromatic radiation is scattered from a surface whose roughness is of the order of a wavelength, causing wave interference which results speckle or multiple-noise in an image, it usually arises in the laser, microscope and synthetic aperture radar (SAR) images. Since the early 1980s, multiple-noise reduction has been a hot topic. ¹ At present, it has been widely studied. The main multiple-noise removal techniques can be divided into four categories: (a) filtering-based methods in spatial domain, (b) transform domain, for example, wavelet domain, (c) non-local filtering, and (d) variational methods. ¹⁷ A typical denoising problem is the combination of tropical algebra with denoising and optimal control. In 2022, Gong et al. ¹⁸ introduced tropical algebra into noise removal, used tropical addition as filter denoising, transformed inequality constraints in traditional algorithms into tropical polynomials, and presented a simple optimization method with constraints given by tropical algebra. Gong pointed out that in the sense of tropical algebra, the traditional optimization problem can be solved by tropical polynomials. The idea provides a baseline and benchmark for accurate algorithms for noise removal and control optimization. Although the multiplicative semigroup of $3\times 3$ tropical matrices has many applications, higher-order tropical matrix multiplicative semigroups have a more extensive and practical prospect. In this paper, the multiplicative semigroup of $4\times 4$ tropical matrices is studied. Next, we first introduce the basic concepts of tropical algebra.

Tropical algebra is the linear algebra of the real numbers augmented with $-\infty$ under the operations of maximum and addition. ¹⁹

$m\oplus n=\mathit{max}\left\{m,n\right\}$

(2)

$m\otimes n=m+n$

(3)

where $m,n\in \mathbb{R}\cup \left\{-\infty \right\}$ and $\mathbb{R}$ is a real number set. For example, we can find,

$\begin{array}{c}3\oplus 7=7,\\ 3\otimes 7=10,\end{array}$

$5\oplus \left(1\otimes 6\right)=\mathit{max}\left\{\mathrm{5,1}+6\right\}=7$

Because of the difference between tropical algebra and traditional algebraic algorithms, tropical algebra has more practical value in mathematics and engineering, and tropical algebra has become the focus of scholars. Tropical algebra has been widely used in combinatorial optimization theory, ²⁰ discrete event system analysis, ²¹ cybernetics, ²² statistical inference, ²³ algebraic geometry, ²⁴ and other disciplines field. Tropical algebra is first used in the field of economic research, with the help of tropical algebra by using the maximum and minimum levels to describe the total profit, this idea can also be used for reference in denoising problems. ²⁵ Tropical addition is a fast and effective filtering algorithm for noise suppression and removal. ²⁶

Tropical algebra has certain characteristics for describing discontinuous functions in optimal control. For example, consider the following polynomial $y\left(x\right)=3x\oplus \left(x\otimes 4\right)$ , after a simple calculation, we get

$y\left(x\right)=3x\oplus \left(x\otimes 4\right)=\mathit{max}\left\{3x,x+4\right\}$

(4)

Equation (4) can be expressed in the traditional way,

$y\left(x\right)=\{\begin{array}{l}3x,x\ge 2\\ x+4,x<2\end{array}$

(5)

Idempotents of tropical matrix semigroups

In 1979, Cuninghame-Green ¹⁹ studied the axiomatization of tropical algebra in detail for the first time. Tropical algebra is the algebraic system $\left(T=\mathbb{R}\cup \left\{-\infty \right\},\oplus ,\otimes \right)$ , where $\oplus$ is the maximum value and $\otimes$ means the ordinary addition. Tropical algebra and many of its basic and important properties have been rediscovered many times. In 2001, Il’in ²⁷ studied the regularity of matrix semigroups over semirings. In 2011, Johnson and Kambites ²⁸ studied multiplicative semigroup of $2\times 2$ tropical matrices, the form of idempotents on this kind of semigroups is given. In 2012, the characterization of $\mathrm{G}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{n}-\mathcal{D}$ relation on multiplicative semigroup of $n\times n$ tropical matrices is given by Hollings and Kambites. ²⁹ In 2013, Johnson and Kambites ³⁰ studied the $\mathrm{G}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{n}-$ J relation on the semigroup of $n\times n$ tropical matrices multiplication. In 2018, the characterization of maximal subgroups containing $n\times n$ tropical nonsingular idempotents $E$ is given by Yang. ³¹ In 2019, Gould et al. ³² gave an equivalent characterization of the Green’s relations of matrix semigroups over semirings. In 2021, Johnson and Kambites ³³ exhibited a faithful representation of the plactic monoid of every finite rank as a monoid of upper triangular matrices over the tropical semiring. In 2022, Thomas ³⁴ studied semigroup varieties generated by full and upper triangular tropical matrix semigroups and the plactic monoid of rank 4. He proved that the upper triangular tropical matrix semigroup $U{T}_n\left(T\right)$ generates a different semigroup variety for each dimension n. In 2022, Kambites ³⁵ studied the free objects in the variety of semigroups and variety of monoids generated by the monoid of all $n\times n$ upper triangular matrices over a commutative semiring. In 2022, Gong et al. ¹⁸ studied idempotents on multiplicative semigroups of $3\times 3$ tropical matrices, and the structure of idempotents on this kind of semigroup is given.

We call $a$ is an idempotent of semigroup $S$ when $a^2=a$ . ³⁶ Two examples are given, we can find $10\oplus 10=10$ and $0\otimes 0=0$ , so 10 is an idempotent of $\left(T,\oplus \right)$ and 0 is an idempotent of $\left(T,\otimes \right)$ . As another example, we can find that,

${\left(\begin{array}{ccc}0& -2& -5\\ -1& 0& -4\\ 2& 3& 0\end{array}\right)}^2=\left(\begin{array}{ccc}0& -2& -5\\ -1& 0& -4\\ 2& 3& 0\end{array}\right)\left(\begin{array}{ccc}0& -2& -5\\ -1& 0& -4\\ 2& 3& 0\end{array}\right)$

$\begin{array}{l}=\left(\begin{array}{ccc}\left(0\otimes 0\right)\oplus \left(-2\otimes -1\right)\oplus \left(-5\otimes 2\right)& \left(0\otimes -2\right)\oplus \left(-2\otimes 0\right)\oplus \left(-5\otimes 3\right)& \left(0\otimes -5\right)\oplus \left(-2\otimes -4\right)\oplus \left(-5\otimes 0\right)\\ \left(-1\otimes 0\right)\oplus \left(0\otimes -1\right)\oplus \left(-4\otimes 2\right)& \left(-1\otimes -2\right)\oplus \left(0\otimes 0\right)\oplus \left(-4\otimes 3\right)& \left(-1\otimes -5\right)\oplus \left(0\otimes -4\right)\oplus \left(-4\otimes 0\right)\\ \left(2\otimes 0\right)\oplus \left(3\otimes -1\right)\oplus \left(0\otimes 2\right)& \left(2\otimes -2\right)\oplus \left(3\otimes 0\right)\oplus \left(0\otimes 3\right)& \left(2\otimes -5\right)\oplus \left(3\otimes -4\right)\oplus \left(0\otimes 0\right)\end{array}\right)\\ =\left(\begin{array}{ccc}0& -2& -5\\ -1& 0& -4\\ 2& 3& 0\end{array}\right).\end{array}$

So $\left(\begin{array}{ccc}0& -2& -5\\ -1& 0& -4\\ 2& 3& 0\end{array}\right)$ is an idempotent of $\left(T,\oplus ,\otimes \right)$ . The idempotent method has become the focus of scholars because it can be easily combined with nonlinear control problems.^37–39 Idempotent analysis deals with functions taking their values in an idempotent semiring and the corresponding function spaces. The idempotent method is a unique method to study the algebraic structure of semigroups. Given idempotent on the multiplicative semigroup of tropical matrices, these semigroups can be recognized and studied more clearly.

Based on this, this paper will study the idempotent classification of the multiplicative semigroup of $4\times 4$ tropical matrices.

Tropical algebra

Definition 1 ²⁸ Let $T=\mathbb{R}\cup \left\{-\infty \right\}$ , define two binary operations $\oplus$ and $\otimes$ on $T$ as follows:

$\left(\forall a,b\in T\right)a\oplus b=\mathit{max}\left\{a,b\right\},a\otimes b=a+b,$

$\left(T,\oplus ,\otimes \right)$ is called tropical algebra. From the tropical algebra, we get:

(i) $\left(T,\oplus \right)$ is a commutative monoid with $-\infty$ as zero element;

(ii) $\left(T,\otimes \right)$ is a commutative monoid with $0$ as identity element;

(iii) For any $a,b,c\in T$ , there are

$\begin{array}{l}a\otimes \left(b\oplus c\right)=\left(a\otimes b\right)\oplus \left(a\otimes c\right),\\ \left(a\oplus b\right)\otimes c=\left(a\otimes c\right)\oplus \left(b\otimes c\right);\end{array}$

• (iv) For any $a,b\in T$ , $a\oplus a=a,a\oplus b=b\oplus a,a\otimes b=b\otimes a$ .

• (v) For any $a\in T$ , $a\otimes \left(-\infty \right)=\left(-\infty \right)\otimes a=-\infty$ .

The above conclusions show that $\left(T,\oplus ,\otimes \right)$ is a commutative semiring, called a tropical semiring. For any $a\in T\mathbb{\backslash }\left\{-\infty \right\}$ , we write $a^{-1}$ as the multiplicative inverse of $a$ , by definition, $a^{-1}$ is actually $-a$ in the traditional sense.

For positive integers $m,n$ , let $M_{m\times n}\left(T\right)$ denote the set of $m\times n$ matrices with entries in $T$ , in particular, when $m=n$ , abbreviated as $M_n\left(T\right)$ (see ⁴⁰ ). The operations of $\oplus$ and $\otimes$ on $T$ can be derived the corresponding operations on $M_n\left(T\right)$ . In fact, for $A,B\in M_n\left(T\right)$ there are

$\begin{array}{l}{\left(A\otimes B\right)}_{i,j}=\stackrel{n}{\underset{k=1}{\oplus }}\left(A_{i,k}\otimes B_{k,j}\right),\\ {\left(A\oplus B\right)}_{i,j}=\left(A_{i,j}\oplus B_{i,j}\right),\end{array}$

For all $1\le i,j\le n$ where $A_{i,j}$ denotes the $\left(i,j\right)\hspace{0.17em}\mathrm{t}\mathrm{h}$ entry of the matrix $A$ . For brevity, we shall usually write $AB$ instead of $A\otimes B$ . Johnson and Kambites ⁴⁰ have pointed out that $M_n\left(T\right)$ is an idempotent semiring, with multiplicative identity,

$I_n=\left(\begin{array}{cccc}0& -\infty & \cdots & -\infty \\ -\infty & 0& \cdots & -\infty \\ ⋮& ⋮& \ddots & ⋮\\ -\infty & -\infty & \dots & 0\end{array}\right)$

And additive identity,

$O=\left(\begin{array}{ccc}-\infty & \cdots & -\infty \\ ⋮& \ddots & ⋮\\ -\infty & \cdots & -\infty \end{array}\right)$

We call $\left(M_n\left(T\right),\oplus ,\otimes \right)$ the $n\times n$ tropical matrix semiring.

Main results

Theorem 1. Let $T$ be a tropical semiring. The idempotents on $M_4\left(T\right)$ are classified as follows:

$\left(\begin{array}{cccc}-\infty & -\infty & -\infty & -\infty \\ -\infty & -\infty & -\infty & -\infty \\ -\infty & -\infty & -\infty & -\infty \\ -\infty & -\infty & -\infty & -\infty \end{array}\right),\left(\begin{array}{cccc}0& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& 0& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& 0& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& 0\end{array}\right),\left(\begin{array}{cccc}0& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& E_{\mathrm{2,1}}{E}_{\mathrm{1,2}}& E_{\mathrm{2,1}}{E}_{\mathrm{1,3}}& E_{\mathrm{2,1}}{E}_{\mathrm{1,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,1}}{E}_{\mathrm{1,2}}& E_{\mathrm{3,1}}{E}_{\mathrm{1,3}}& E_{\mathrm{3,1}}{E}_{\mathrm{1,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,1}}{E}_{\mathrm{1,2}}& E_{\mathrm{4,1}}{E}_{\mathrm{1,3}}& E_{\mathrm{4,1}}{E}_{\mathrm{1,4}}\end{array}\right)$

$\left(\begin{array}{cccc}{E}_{\mathrm{1,2}}{E}_{\mathrm{2,1}}& E_{\mathrm{1,2}}& E_{\mathrm{1,2}}{E}_{\mathrm{2,3}}& E_{\mathrm{1,2}}{E}_{\mathrm{2,4}}\\ E_{\mathrm{2,1}}& 0& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,2}}{E}_{\mathrm{2,1}}& E_{\mathrm{3,2}}& E_{\mathrm{3,2}}{E}_{\mathrm{2,3}}& E_{\mathrm{3,2}}{E}_{\mathrm{2,4}}\\ E_{\mathrm{4,2}}{E}_{\mathrm{2,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,2}}{E}_{\mathrm{2,3}}& E_{\mathrm{4,2}}{E}_{\mathrm{2,4}}\end{array}\right),\left(\begin{array}{cccc}{E}_{\mathrm{1,3}}{E}_{\mathrm{3,1}}& E_{\mathrm{1,3}}{E}_{\mathrm{3,2}}& E_{13}& E_{\mathrm{1,3}}{E}_{\mathrm{3,4}}\\ E_{\mathrm{2,3}}{E}_{\mathrm{3,1}}& E_{\mathrm{2,3}}{E}_{\mathrm{3,2}}& E_{\mathrm{2,3}}& E_{\mathrm{2,3}}{E}_{\mathrm{3,4}}\\ E_{\mathrm{3,3}}{E}_{\mathrm{3,1}}& E_{\mathrm{3,2}}& 0& E_{\mathrm{3,4}}\\ E_{\mathrm{4,3}}{E}_{\mathrm{3,1}}& E_{\mathrm{4,3}}{E}_{\mathrm{3,4}}& E_{\mathrm{4,3}}& E_{\mathrm{4,3}}{E}_{\mathrm{3,4}}\end{array}\right),$

$\left(\begin{array}{cccc}{E}_{\mathrm{1,4}}{E}_{\mathrm{4,1}}& E_{\mathrm{1,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{1,4}}{E}_{\mathrm{4,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,4}}{E}_{\mathrm{4,1}}& E_{\mathrm{2,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{2,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,4}}{E}_{\mathrm{4,1}}& E_{\mathrm{3,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{3,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& 0\end{array}\right),\left(\begin{array}{cccc}0& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& 0& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& E_{\mathrm{3,1}}{E}_{\mathrm{1,3}}\oplus E_{\mathrm{3,2}}{E}_{\mathrm{2,3}}& E_{\mathrm{3,1}}{E}_{\mathrm{1,4}}\oplus E_{\mathrm{3,2}}{E}_{\mathrm{2,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,1}}{E}_{\mathrm{1,3}}\oplus E_{\mathrm{4,2}}{E}_{\mathrm{2,3}}& E_{\mathrm{4,1}}{E}_{\mathrm{1,4}}\oplus E_{\mathrm{4,2}}{E}_{\mathrm{2,4}}\end{array}\right),$

$\left(\begin{array}{cccc}0& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& E_{\mathrm{2,1}}{E}_{\mathrm{1,2}}\oplus E_{\mathrm{2,3}}{E}_{\mathrm{3,2}}& E_{\mathrm{2,3}}& E_{\mathrm{2,1}}{E}_{\mathrm{1,4}}\oplus E_{\mathrm{2,3}}{E}_{\mathrm{3,4}}\\ E_{\mathrm{3,1}}& A_{\mathrm{3,2}}& 0& A_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,1}}{E}_{\mathrm{1,2}}\oplus E_{\mathrm{4,3}}{E}_{\mathrm{3,2}}& E_{\mathrm{4,3}}& E_{\mathrm{4,1}}{E}_{\mathrm{1,4}}\oplus E_{\mathrm{4,3}}{E}_{\mathrm{3,4}}\end{array}\right),\left(\begin{array}{cccc}0& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& E_{\mathrm{2,1}}{E}_{\mathrm{1,2}}\oplus E_{\mathrm{2,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{2,1}}{E}_{\mathrm{1,3}}\oplus E_{\mathrm{2,4}}{E}_{\mathrm{4,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,1}}{E}_{\mathrm{1,2}}\oplus E_{\mathrm{3,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{3,1}}{E}_{\mathrm{1,3}}\oplus E_{\mathrm{3,4}}{E}_{\mathrm{4,3}}& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& 0\end{array}\right),$

$\left(\begin{array}{cccc}{E}_{\mathrm{1,2}}{E}_{\mathrm{2,1}}\oplus E_{\mathrm{1,3}}{E}_{\mathrm{3,1}}& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,2}}{E}_{\mathrm{2,4}}\oplus E_{\mathrm{1,3}}{E}_{\mathrm{3,4}}\\ E_{\mathrm{2,1}}& 0& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& 0& E_{\mathrm{3,4}}\\ E_{\mathrm{4,2}}{E}_{\mathrm{2,1}}\oplus E_{\mathrm{4,3}}{E}_{\mathrm{3,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& E_{\mathrm{4,2}}{E}_{\mathrm{2,4}}\oplus E_{\mathrm{4,3}}{E}_{\mathrm{3,4}}\end{array}\right),\left(\begin{array}{cccc}{E}_{\mathrm{1,2}}{E}_{\mathrm{2,1}}\oplus E_{\mathrm{1,4}}{E}_{\mathrm{4,1}}& E_{\mathrm{1,2}}& E_{\mathrm{1,2}}{E}_{\mathrm{2,3}}\oplus E_{\mathrm{1,4}}{E}_{\mathrm{4,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& 0& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,2}}{E}_{\mathrm{2,1}}\oplus E_{\mathrm{3,4}}{E}_{\mathrm{4,1}}& E_{\mathrm{3,2}}& E_{\mathrm{3,2}}{E}_{\mathrm{2,3}}\oplus E_{\mathrm{3,4}}{E}_{\mathrm{4,3}}& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& 0\end{array}\right),$

$\left(\begin{array}{cccc}{E}_{\mathrm{1,3}}{E}_{\mathrm{3,1}}\oplus E_{\mathrm{1,4}}{E}_{\mathrm{4,1}}& E_{\mathrm{1,3}}{E}_{\mathrm{3,2}}\oplus E_{\mathrm{1,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,3}}{E}_{\mathrm{3,1}}\oplus E_{\mathrm{2,4}}{E}_{\mathrm{4,1}}& E_{\mathrm{2,3}}{E}_{\mathrm{3,2}}\oplus E_{\mathrm{2,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& 0& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& 0\end{array}\right),\left(\begin{array}{cccc}0& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& 0& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& E_{\mathrm{3,1}}{E}_{\mathrm{1,3}}\oplus E_{\mathrm{3,2}}{E}_{\mathrm{2,3}}\oplus E_{\mathrm{3,4}}{E}_{\mathrm{4,3}}& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& 0\end{array}\right),$

$\left(\begin{array}{cccc}0& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& 0& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& 0& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& E_{\mathrm{4,1}}{E}_{\mathrm{1,4}}\oplus E_{\mathrm{4,2}}{E}_{\mathrm{2,4}}\oplus E_{\mathrm{4,3}}{E}_{\mathrm{3,4}}\end{array}\right),\left(\begin{array}{cccc}0& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& E_{\mathrm{2,1}}{E}_{\mathrm{1,2}}\oplus E_{\mathrm{2,3}}{E}_{\mathrm{3,2}}\oplus E_{\mathrm{2,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& 0& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& 0\end{array}\right),$

$\begin{array}{l}\left(\begin{array}{cccc}{E}_{\mathrm{1,2}}{E}_{\mathrm{2,1}}\oplus E_{\mathrm{1,3}}{E}_{\mathrm{3,1}}\oplus E_{\mathrm{1,4}}{E}_{\mathrm{4,1}}& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& 0& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& 0& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& 0\end{array}\right),\end{array}$

$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} E_{i,j}\in T \mathrm{a}\mathrm{n}\mathrm{d} E_{i,j}{E}_{j,i}\le 0,E_{i,k}{E}_{k,j}\le E_{i,j}\left(i\ne j\right)$

ProofLet $E=\left(\begin{array}{cccc}{E}_{\mathrm{1,1}}& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& E_{\mathrm{2,2}}& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& E_{\mathrm{3,3}}& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& E_{\mathrm{4,4}}\end{array}\right)$ , then

$E^2=\left(\begin{array}{cccc}\stackrel{4}{\underset{k=1}{\oplus }}\left(E_{1,k}\otimes E_{k,1}\right)& \stackrel{4}{\underset{k=1}{\oplus }}\left(E_{1,k}\otimes E_{k,2}\right)& \stackrel{4}{\underset{k=1}{\oplus }}\left(A_{1,k}\otimes E_{k,3}\right)& \stackrel{4}{\underset{k=1}{\oplus }}\left(E_{1,k}\otimes E_{k,4}\right)\\ \stackrel{4}{\underset{k=1}{\oplus }}\left(E_{2,k}\otimes E_{k,1}\right)& \stackrel{4}{\underset{k=1}{\oplus }}\left(E_{2,k}\otimes E_{k,2}\right)& \stackrel{4}{\underset{k=1}{\oplus }}\left(E_{2,k}\otimes E_{k,3}\right)& \stackrel{4}{\underset{k=1}{\oplus }}\left(E_{2,k}\otimes E_{k,4}\right)\\ \stackrel{4}{\underset{k=1}{\oplus }}\left(E_{3,k}\otimes E_{k,1}\right)& \stackrel{4}{\underset{k=1}{\oplus }}\left(E_{3,k}\otimes E_{k,2}\right)& \stackrel{4}{\underset{k=1}{\oplus }}\left(E_{3,k}\otimes E_{k,3}\right)& \stackrel{4}{\underset{k=1}{\oplus }}\left(E_{3,k}\otimes E_{k,4}\right)\\ \stackrel{4}{\underset{k=1}{\oplus }}\left(E_{4,k}\otimes E_{k,1}\right)& \stackrel{4}{\underset{k=1}{\oplus }}\left(E_{4,k}\otimes E_{k,2}\right)& \stackrel{4}{\underset{k=1}{\oplus }}\left(E_{4,k}\otimes E_{k,3}\right)& \stackrel{4}{\underset{k=1}{\oplus }}\left(E_{4,k}\otimes E_{k,4}\right)\end{array}\right)$

So,

$E_{\mathrm{1,1}}=\mathit{max}\left\{E_{\mathrm{1,1}}+E_{\mathrm{1,1}},E_{\mathrm{1,2}}+E_{\mathrm{2,1}},E_{\mathrm{1,3}}+E_{\mathrm{3,1}},E_{\mathrm{1,4}}+E_{\mathrm{4,1}}\right\}$

(6)

$E_{\mathrm{1,2}}=\mathit{max}\left\{E_{\mathrm{1,1}}+E_{\mathrm{1,2}},E_{\mathrm{1,2}}+E_{\mathrm{2,2}},E_{\mathrm{1,3}}+E_{\mathrm{3,2}},E_{\mathrm{1,4}}+E_{\mathrm{4,2}}\right\}$

(7)

$E_{\mathrm{1,3}}=\mathit{max}\left\{E_{\mathrm{1,1}}+E_{\mathrm{1,3}},E_{\mathrm{1,2}}+E_{\mathrm{2,3}},E_{\mathrm{1,3}}+E_{\mathrm{3,3}},E_{\mathrm{1,4}}+E_{\mathrm{4,3}}\right\}$

(8)

$E_{\mathrm{1,4}}=\mathit{max}\left\{E_{\mathrm{1,1}}+E_{\mathrm{1,4}},E_{\mathrm{1,2}}+E_{\mathrm{2,4}},E_{\mathrm{1,3}}+E_{\mathrm{3,4}},E_{\mathrm{1,4}}+E_{\mathrm{4,4}}\right\}$

(9)

$E_{\mathrm{2,1}}=\mathit{max}\left\{E_{\mathrm{2,1}}+E_{\mathrm{1,1}},E_{\mathrm{2,2}}+E_{\mathrm{2,1}},E_{\mathrm{2,3}}+E_{\mathrm{3,1}},E_{\mathrm{2,4}}+E_{\mathrm{4,1}}\right\}$

(10)

$E_{\mathrm{2,2}}=\mathit{max}\left\{E_{\mathrm{2,1}}+E_{\mathrm{1,2}},E_{\mathrm{2,2}}+E_{\mathrm{2,2}},E_{\mathrm{2,3}}+E_{\mathrm{3,2}},E_{\mathrm{2,4}}+E_{\mathrm{4,2}}\right\}$

(11)

$E_{\mathrm{2,3}}=\mathit{max}\left\{E_{\mathrm{2,1}}+E_{\mathrm{1,3}},E_{\mathrm{2,2}}+E_{\mathrm{2,3}},E_{\mathrm{2,3}}+E_{\mathrm{3,3}},E_{\mathrm{2,4}}+E_{\mathrm{4,3}}\right\}$

(12)

$E_{\mathrm{2,4}}=\mathit{max}\left\{E_{\mathrm{2,1}}+E_{\mathrm{1,4}},E_{\mathrm{2,2}}+E_{\mathrm{2,4}},E_{\mathrm{2,3}}+E_{\mathrm{3,4}},E_{\mathrm{2,4}}+E_{\mathrm{4,4}}\right\}$

(13)

$E_{\mathrm{3,1}}=\mathit{max}\left\{E_{\mathrm{3,1}}+E_{\mathrm{1,1}},E_{\mathrm{3,2}}+E_{\mathrm{2,1}},E_{\mathrm{3,3}}+E_{\mathrm{3,1}},E_{\mathrm{3,4}}+E_{\mathrm{4,1}}\right\}$

(14)

${\mathit{E}}_{\mathrm{3,2}}=\mathit{max}\left\{{\mathrm{E}}_{\mathrm{3,1}}+{\mathrm{E}}_{\mathrm{1,2}},{\mathrm{E}}_{\mathrm{3,2}}+{\mathrm{E}}_{\mathrm{2,2}},{\mathrm{E}}_{\mathrm{3,3}}+{\mathrm{E}}_{\mathrm{3,2}},{\mathrm{E}}_{\mathrm{3,4}}+{\mathrm{E}}_{\mathrm{4,2}}\right\}$

(15)

${\mathit{E}}_{\mathrm{3,3}}=\mathit{max}\left\{{\mathrm{E}}_{\mathrm{3,1}}+{\mathrm{E}}_{\mathrm{1,3}},{\mathrm{E}}_{\mathrm{3,2}}+{\mathrm{E}}_{\mathrm{2,3}},{\mathrm{E}}_{\mathrm{3,3}}+{\mathrm{E}}_{\mathrm{3,3}},{\mathrm{E}}_{\mathrm{3,4}}+{\mathrm{E}}_{\mathrm{4,3}}\right\}$

(16)

${\mathit{E}}_{\mathrm{3,4}}=\mathit{max}\left\{{\mathrm{E}}_{\mathrm{3,1}}+{\mathrm{E}}_{\mathrm{1,4}},{\mathrm{E}}_{\mathrm{3,2}}+{\mathrm{E}}_{\mathrm{2,4}},{\mathrm{E}}_{\mathrm{3,3}}+{\mathrm{E}}_{\mathrm{3,4}},{\mathrm{E}}_{\mathrm{3,4}}+{\mathrm{E}}_{\mathrm{4,4}}\right\}$

(17)

${\mathit{E}}_{\mathrm{4,1}}=\mathit{max}\left\{{\mathrm{E}}_{\mathrm{4,1}}+{\mathrm{E}}_{\mathrm{1,1}},{\mathrm{E}}_{\mathrm{4,2}}+{\mathrm{E}}_{\mathrm{2,1}},{\mathrm{E}}_{\mathrm{4,3}}+{\mathrm{E}}_{\mathrm{3,1}},{\mathrm{E}}_{\mathrm{4,4}}+{\mathrm{E}}_{\mathrm{4,1}}\right\}$

(18)

${\mathit{E}}_{\mathrm{4,2}}=\mathit{max}\left\{{\mathrm{E}}_{\mathrm{4,1}}+{\mathrm{E}}_{\mathrm{1,2}},{\mathrm{E}}_{\mathrm{4,2}}+{\mathrm{E}}_{\mathrm{2,2}},{\mathrm{E}}_{\mathrm{4,3}}+{\mathrm{E}}_{\mathrm{3,2}},{\mathrm{E}}_{\mathrm{4,4}}+{\mathrm{E}}_{\mathrm{4,2}}\right\}$

(19)

${\mathit{E}}_{\mathrm{4,3}}=\mathit{max}\left\{{\mathrm{E}}_{\mathrm{4,1}}+{\mathrm{E}}_{\mathrm{1,3}},{\mathrm{E}}_{\mathrm{4,2}}+{\mathrm{E}}_{\mathrm{2,3}},{\mathrm{E}}_{\mathrm{4,3}}+{\mathrm{E}}_{\mathrm{3,3}},{\mathrm{E}}_{\mathrm{4,4}}+{\mathrm{E}}_{\mathrm{4,3}}\right\}$

(20)

${\mathit{E}}_{\mathrm{4,4}}=\mathit{max}\left\{{\mathrm{E}}_{\mathrm{4,1}}+{\mathrm{E}}_{\mathrm{1,4}},{\mathrm{E}}_{\mathrm{4,2}}+{\mathrm{E}}_{\mathrm{2,4}},{\mathrm{E}}_{\mathrm{4,3}}+{\mathrm{E}}_{\mathrm{3,4}},{\mathrm{E}}_{\mathrm{4,4}}+{\mathrm{E}}_{\mathrm{4,4}}\right\}$

(21)

From equations (6), (11), (16) and (21) we have $E_{\mathrm{1,1}}+E_{\mathrm{1,1}}\le E_{\mathrm{1,1}},E_{\mathrm{2,2}}+E_{\mathrm{2,2}}\le E_{\mathrm{2,2}},E_{\mathrm{3,3}}+E_{\mathrm{3,3}}\le E_{\mathrm{3,3}},E_{\mathrm{4,4}}+E_{\mathrm{4,4}}\le E_{\mathrm{4,4}}$ .

So $E_{\mathrm{1,1}}\le 0,E_{\mathrm{2,2}}\le 0,E_{\mathrm{3,3}}\le 0,E_{\mathrm{4,4}}\le 0.$ Thus, there are sixteen situations as follows:

Situation One: $E_{\mathrm{1,1}}<0,E_{\mathrm{2,2}}<0,E_{\mathrm{3,3}}<0,E_{\mathrm{4,4}}<0.$ .

From equation (7), we have $E_{\mathrm{1,2}}=\mathit{max}\left\{E_{\mathrm{1,3}}+E_{\mathrm{3,2}},E_{\mathrm{1,4}}+E_{\mathrm{4,2}}\right\}$ . If $E_{\mathrm{1,2}}=E_{\mathrm{1,3}}+E_{\mathrm{3,2}}$ , from equation (8) we can know that there are three situations:

(a) $E_{\mathrm{1,2}}=E_{\mathrm{1,2}}+E_{\mathrm{2,3}}+E_{\mathrm{3,2}}$ , from equation (11) we have $E_{\mathrm{2,3}}+E_{\mathrm{3,2}}\le E_{\mathrm{2,2}}<0$ , so $E_{\mathrm{1,2}}=-\infty$ .

(b) $E_{\mathrm{1,2}}=E_{\mathrm{1,4}}+E_{\mathrm{4,3}}+E_{\mathrm{3,1}}+E_{\mathrm{1,2}}$ , from equations (8) and (10) we get $E_{\mathrm{1,4}}+E_{\mathrm{4,3}}+E_{\mathrm{3,1}}\le E_{\mathrm{1,3}}+E_{\mathrm{3,1}}<0$ , so $E_{\mathrm{1,2}}=-\infty$ .

(c) $E_{\mathrm{1,2}}=E_{\mathrm{1,4}}+E_{\mathrm{4,3}}+E_{\mathrm{3,4}}+E_{\mathrm{4,2}}$ it is known from the equation (19) that there are two situations at this time:

(i) If $E_{\mathrm{4,2}}=E_{\mathrm{4,1}}+E_{\mathrm{1,2}}$ , then $E_{\mathrm{1,2}}=E_{\mathrm{1,4}}+E_{\mathrm{4,3}}+E_{\mathrm{3,4}}+E_{\mathrm{4,1}}+E_{\mathrm{1,2}}$ , from equations (6) and (11) we have $E_{\mathrm{1,4}}+E_{\mathrm{4,1}}<0,E_{\mathrm{4,3}}+E_{\mathrm{3,4}}<0,$ so $E_{\mathrm{1,2}}=-\infty$ .

(ii) If $E_{\mathrm{4,2}}=E_{\mathrm{4,3}}+E_{\mathrm{3,2}}$ , from $E_{\mathrm{3,2}}=E_{\mathrm{3,4}}+E_{\mathrm{4,2}}$ and equation (21) we obtain $E_{\mathrm{4,2}}=E_{\mathrm{4,3}}+E_{\mathrm{3,4}}+E_{\mathrm{4,2}}\le E_{\mathrm{4,4}}+E_{\mathrm{4,2}}$ , then $E_{\mathrm{4,2}}=-\infty$ , so $E_{\mathrm{1,2}}=-\infty$ .

If $E_{\mathrm{1,2}}=E_{\mathrm{1,4}}+E_{\mathrm{4,2}}$ , then $E_{\mathrm{1,2}}=-\infty$ can be obtained by similar discussion. In the same way, $E_{\mathrm{1,3}}=E_{\mathrm{1,4}}=E_{\mathrm{2,1}}=E_{\mathrm{2,3}}=E_{\mathrm{2,4}}=E_{\mathrm{3,1}}=E_{\mathrm{3,2}}=E_{\mathrm{3,4}}=E_{\mathrm{4,1}}=E_{\mathrm{4,2}}=E_{\mathrm{4,3}}=-\infty$ . And then we get $E_{\mathrm{1,1}}=E_{\mathrm{2,2}}=E_{\mathrm{3,3}}=E_{\mathrm{4,4}}=-\infty .$ So, we have,

$E=\left(\begin{array}{cccc}-\infty & -\infty & -\infty & -\infty \\ -\infty & -\infty & -\infty & -\infty \\ -\infty & -\infty & -\infty & -\infty \\ -\infty & -\infty & -\infty & -\infty \end{array}\right).$

Situation Two: $E_{\mathrm{1,1}}=0{, E}_{\mathrm{2,2}}<0{, E}_{\mathrm{3,3}}<0{,E}_{\mathrm{4,4}}<0.$

From equation (13) we know $E_{\mathrm{2,4}}=\mathit{max}\left\{E_{\mathrm{2,1}}+E_{\mathrm{1,4}},E_{\mathrm{2,3}}+E_{\mathrm{3,4}}\right\}$ . Let $E_{\mathrm{2,4}}=E_{\mathrm{2,3}}+E_{\mathrm{3,4}}$ , then $E_{\mathrm{2,1}}+E_{\mathrm{1,4}}\le E_{\mathrm{2,3}}+E_{\mathrm{3,4}}$ . From equation (12) we get $E_{\mathrm{2,3}}=\mathit{max}\left\{E_{\mathrm{2,1}}+E_{\mathrm{1,3}},E_{\mathrm{2,4}}+E_{\mathrm{4,3}}\right\}$ and then $E_{\mathrm{2,3}}=\mathit{max}\left\{E_{\mathrm{2,1}}+E_{\mathrm{1,3}},E_{\mathrm{2,3}}+E_{\mathrm{3,4}}+E_{\mathrm{4,3}}\right\}$ . By equation (16) we get $E_{\mathrm{3,4}}+E_{\mathrm{4,3}}\le E_{\mathrm{3,3}}< 0$ , so $E_{\mathrm{2,3}}=E_{\mathrm{2,1}}+E_{\mathrm{1,3}}$ . By equation (17) we know $E_{\mathrm{3,4}}=\mathit{max}\left\{E_{\mathrm{3,1}}+E_{\mathrm{1,4}},E_{\mathrm{3,2}}+E_{\mathrm{2,4}}\right\}$ and then $E_{\mathrm{3,4}}=\mathit{max}\left\{E_{\mathrm{3,1}}+E_{\mathrm{1,4}},E_{\mathrm{3,2}}+E_{\mathrm{2,3}}+E_{\mathrm{3,4}}\right\}$ . By equation (16) we obtain $E_{\mathrm{3,4}}+E_{\mathrm{4,3}}\le E_{\mathrm{3,3}}< 0$ , so $E_{\mathrm{3,4}}=E_{\mathrm{3,1}}+E_{\mathrm{1,4}}$ , thus,

$\begin{array}{l}{E}_{\mathrm{2,3}}+E_{\mathrm{3,4}}=\left(E_{\mathrm{2,1}}+E_{\mathrm{1,3}}\right)+\left(E_{\mathrm{3,1}}+E_{\mathrm{1,4}}\right)\\ =\left(E_{\mathrm{2,1}}+E_{\mathrm{1,4}}\right)+\left(E_{\mathrm{1,3}}+E_{\mathrm{3,1}}\right)\\ \le E_{\mathrm{2,1}}+E_{\mathrm{1,4}}.\end{array}$

Therefore, $E_{\mathrm{2,1}}+E_{\mathrm{1,4}}=E_{\mathrm{2,3}}+E_{\mathrm{3,4}}$ , so $E_{\mathrm{2,4}}=E_{\mathrm{2,1}}+E_{\mathrm{1,4}}$ . In the same way, we can get $E_{\mathrm{2,3}}=E_{\mathrm{2,1}}+E_{\mathrm{1,3}},E_{\mathrm{2,4}}=E_{\mathrm{2,1}}+E_{\mathrm{1,4}},E_{\mathrm{3,2}}=E_{\mathrm{3,1}}+E_{\mathrm{1,2}},E_{\mathrm{3,4}}=E_{\mathrm{3,1}}+E_{\mathrm{1,4}},E_{\mathrm{4,2}}=E_{\mathrm{4,1}}+E_{\mathrm{1,2}},E_{\mathrm{4,3}}=E_{\mathrm{4,1}}+E_{\mathrm{1,3}}.$ It can be obtained $E_{\mathrm{1,3}}+E_{\mathrm{3,1}}\le A_{\mathrm{1,1}}=0,E_{\mathrm{1,4}}+E_{\mathrm{4,1}}\le E_{\mathrm{1,1}}=0$ in the same way. Thus,

$\begin{array}{l}{E}_{\mathrm{2,2}}=\mathit{max}\left\{E_{\mathrm{2,1}}+E_{\mathrm{1,2}},E_{\mathrm{2,3}}+E_{\mathrm{3,2}},E_{\mathrm{2,4}}+E_{\mathrm{4,2}}\right\}\\ =\mathit{max}\left\{E_{\mathrm{2,1}}+E_{\mathrm{1,2}},E_{\mathrm{2,1}}+E_{\mathrm{1,3}}+E_{\mathrm{3,1}}+E_{\mathrm{1,2}},E_{\mathrm{2,1}}+E_{\mathrm{1,4}}+E_{\mathrm{4,1}}+E_{\mathrm{1,2}}\right\}\\ =E_{\mathrm{2,1}}+E_{\mathrm{1,2}}.\end{array}$

It can be obtained $E_{\mathrm{3,3}}=E_{\mathrm{3,1}}+E_{\mathrm{1,3}},E_{\mathrm{4,4}}=E_{\mathrm{4,1}}+E_{\mathrm{1,4}}$ in the same way. So, we have

$E=\left(\begin{array}{cccc}0& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& E_{\mathrm{2,1}}{E}_{\mathrm{1,2}}& E_{\mathrm{2,1}}{E}_{\mathrm{1,3}}& E_{\mathrm{2,1}}{E}_{\mathrm{1,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,1}}{E}_{\mathrm{1,2}}& E_{\mathrm{3,1}}{E}_{\mathrm{1,3}}& E_{\mathrm{3,1}}{E}_{\mathrm{1,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,1}}{E}_{\mathrm{1,2}}& E_{\mathrm{4,1}}{E}_{\mathrm{1,3}}& E_{\mathrm{4,1}}{E}_{\mathrm{1,4}}\end{array}\right)\mathrm{a}\mathrm{n}\mathrm{d} E_{i,k}{E}_{k,j}\le E_{i,j}\left(i\ne j\right).$

Situation Three: $E_{\mathrm{1,1}}<0{, E}_{\mathrm{2,2}}=0{, E}_{\mathrm{3,3}}<0{,E}_{\mathrm{4,4}}<0.$

Similar Situation Two, we have,

$E=\left(\begin{array}{cccc}{E}_{\mathrm{1,2}}{E}_{\mathrm{2,1}}& E_{\mathrm{1,2}}& E_{\mathrm{1,2}}{E}_{\mathrm{2,3}}& E_{\mathrm{1,2}}{E}_{\mathrm{2,4}}\\ E_{\mathrm{2,1}}& 0& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,2}}{E}_{\mathrm{2,1}}& E_{\mathrm{3,2}}& E_{\mathrm{3,2}}{E}_{\mathrm{2,3}}& E_{\mathrm{3,2}}{E}_{\mathrm{2,4}}\\ E_{\mathrm{4,2}}{E}_{\mathrm{2,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,2}}{E}_{\mathrm{2,3}}& E_{\mathrm{4,2}}{E}_{\mathrm{2,4}}\end{array}\right) \mathrm{a}\mathrm{n}\mathrm{d} E_{i,k}{E}_{k,j}\le E_{i,j}\left(i\ne j\right).$

Situation Four: $E_{\mathrm{1,1}}<0{, E}_{\mathrm{2,2}}<0{, E}_{\mathrm{3,3}}=0{,E}_{\mathrm{4,4}}<0.$

Similar Situation Two, we have,

$E=\left(\begin{array}{cccc}{E}_{\mathrm{1,3}}{E}_{\mathrm{3,1}}& E_{\mathrm{1,3}}{E}_{\mathrm{3,2}}& E_{13}& E_{\mathrm{1,3}}{E}_{\mathrm{3,4}}\\ E_{\mathrm{2,3}}{E}_{\mathrm{3,1}}& E_{\mathrm{2,3}}{E}_{\mathrm{3,2}}& E_{\mathrm{2,3}}& E_{\mathrm{2,3}}{E}_{\mathrm{3,4}}\\ E_{\mathrm{3,3}}{E}_{\mathrm{3,1}}& E_{\mathrm{3,2}}& 0& E_{\mathrm{3,4}}\\ E_{\mathrm{4,3}}{E}_{\mathrm{3,1}}& E_{\mathrm{4,3}}{E}_{\mathrm{3,4}}& E_{\mathrm{4,3}}& E_{\mathrm{4,3}}{E}_{\mathrm{3,4}}\end{array}\right) \mathrm{a}\mathrm{n}\mathrm{d} E_{i,k}{E}_{k,j}\le E_{i,j}\left(i\ne j\right).$

Situation Five: $E_{\mathrm{1,1}}<0{, E}_{\mathrm{2,2}}<0{, E}_{\mathrm{3,3}}<0{,E}_{\mathrm{4,4}}=0.$

Similar Situation Two, we have,

$E=\left(\begin{array}{cccc}{E}_{\mathrm{1,4}}{E}_{\mathrm{4,1}}& E_{\mathrm{1,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{1,4}}{E}_{\mathrm{4,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,4}}{E}_{\mathrm{4,1}}& E_{\mathrm{2,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{2,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,4}}{E}_{\mathrm{4,1}}& E_{\mathrm{3,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{3,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& 0\end{array}\right)\mathrm{a}\mathrm{n}\mathrm{d} E_{i,k}{E}_{k,j}\le E_{i,j}\left(i\ne j\right).$

Situation Six: $E_{\mathrm{1,1}}{=E}_{\mathrm{2,2}}=0{, E}_{\mathrm{3,3}}<0{,E}_{\mathrm{4,4}}<0.$

From equations (17) and (20), we obtain $E_{\mathrm{3,4}}=\mathit{max}\left\{E_{\mathrm{3,1}}+E_{\mathrm{1,4}},E_{\mathrm{3,2}}+E_{\mathrm{2,4}}\right\},$ $E_{\mathrm{4,3}}=\mathit{max}\left\{E_{\mathrm{4,1}}+E_{\mathrm{1,3}},E_{\mathrm{4,2}}+E_{\mathrm{2,3}}\right\}$ , and then,

$\begin{array}{c}{E}_{\mathrm{4,3}}=\mathit{max}\{E_{\mathrm{3,1}}+E_{\mathrm{1,4}}+E_{\mathrm{4,1}}+E_{\mathrm{1,3}},E_{\mathrm{3,1}}+E_{\mathrm{1,4}}+E_{\mathrm{4,2}}+E_{\mathrm{2,3}},\\ \hspace{1em}{E}_{\mathrm{3,2}}+E_{\mathrm{2,4}}+E_{\mathrm{4,1}}+E_{\mathrm{1,3}},E_{\mathrm{3,2}}+E_{\mathrm{2,4}}+E_{\mathrm{4,2}}+E_{\mathrm{2,3}}\}.\end{array}$

From equation (6), we know $E_{\mathrm{1,4}}{E}_{\mathrm{4,1}}\le E_{\mathrm{1,1}}=0$ , from equations (7) and (8) we have $E_{\mathrm{1,4}}{E}_{\mathrm{4,2}}{E}_{\mathrm{2,3}}\le E_{\mathrm{1,3}}$ , by equations (10) and (12) we obtain $E_{\mathrm{2,4}}{E}_{\mathrm{4,1}}{E}_{\mathrm{1,3}}\le E_{\mathrm{2,3}}$ , by equation (11) we have $E_{\mathrm{2,4}}{E}_{\mathrm{4,2}}\le E_{\mathrm{2,2}}=0$ . So $E_{\mathrm{3,3}}=\mathit{max}\left\{E_{\mathrm{3,1}}+E_{\mathrm{1,3}},E_{\mathrm{3,2}}+E_{\mathrm{2,3}}\right\}.$ In the same way, we can get $E_{\mathrm{4,4}}=\mathit{max}\left\{E_{\mathrm{4,1}}+E_{\mathrm{1,4}},E_{\mathrm{4,2}}+E_{\mathrm{2,4}}\right\}$ . So, we have,

$E=\left(\begin{array}{cccc}0& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& 0& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& E_{\mathrm{3,1}}{E}_{\mathrm{1,3}}\oplus E_{\mathrm{3,2}}{E}_{\mathrm{2,3}}& E_{\mathrm{3,1}}{E}_{\mathrm{1,4}}\oplus E_{\mathrm{3,2}}{E}_{\mathrm{2,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,1}}{E}_{\mathrm{1,3}}\oplus E_{\mathrm{4,2}}{E}_{\mathrm{2,3}}& E_{\mathrm{4,1}}{E}_{\mathrm{1,4}}\oplus E_{\mathrm{4,2}}{E}_{\mathrm{2,4}}\end{array}\right) \mathrm{a}\mathrm{n}\mathrm{d} E_{i,k}{E}_{k,j}\le E_{i,j}\left(i\ne j\right).$

Situation Seven: $E_{\mathrm{1,1}}{=E}_{\mathrm{3,3}}=0{, E}_{\mathrm{2,2}}<0{,E}_{\mathrm{4,4}}<0.$

Similar Situation Six, we have,

$E=\left(\begin{array}{cccc}0& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& E_{\mathrm{2,1}}{E}_{\mathrm{1,2}}\oplus E_{\mathrm{2,3}}{E}_{\mathrm{3,2}}& E_{\mathrm{2,3}}& E_{\mathrm{2,1}}{E}_{\mathrm{1,4}}\oplus E_{\mathrm{2,3}}{E}_{\mathrm{3,4}}\\ E_{\mathrm{3,1}}& A_{\mathrm{3,2}}& 0& A_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,1}}{E}_{\mathrm{1,2}}\oplus E_{\mathrm{4,3}}{E}_{\mathrm{3,2}}& E_{\mathrm{4,3}}& E_{\mathrm{4,1}}{E}_{\mathrm{1,4}}\oplus E_{\mathrm{4,3}}{E}_{\mathrm{3,4}}\end{array}\right) \mathrm{a}\mathrm{n}\mathrm{d} E_{i,k}{E}_{k,j}\le E_{i,j}\left(i\ne j\right).$

Situation Eight: $E_{\mathrm{1,1}}{=E}_{\mathrm{4,4}}=0{, E}_{\mathrm{2,2}}<0{,E}_{\mathrm{3,3}}<0.$

Similar Situation Six, we have,

$E=\left(\begin{array}{cccc}0& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& E_{\mathrm{2,1}}{E}_{\mathrm{1,2}}\oplus E_{\mathrm{2,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{2,1}}{E}_{\mathrm{1,3}}\oplus E_{\mathrm{2,4}}{E}_{\mathrm{4,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,1}}{E}_{\mathrm{1,2}}\oplus E_{\mathrm{3,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{3,1}}{E}_{\mathrm{1,3}}\oplus E_{\mathrm{3,4}}{E}_{\mathrm{4,3}}& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& 0\end{array}\right) \mathrm{a}\mathrm{n}\mathrm{d} E_{i,k}{E}_{k,j}\le E_{i,j}\left(i\ne j\right).$

Situation Nine: $E_{\mathrm{2,2}}{=E}_{\mathrm{3,3}}=0{, E}_{\mathrm{1,1}}<0{,E}_{\mathrm{4,4}}<0.$

Similar Situation Six, we have,

$E=\left(\begin{array}{cccc}{E}_{\mathrm{1,2}}{E}_{\mathrm{2,1}}\oplus E_{\mathrm{1,3}}{E}_{\mathrm{3,1}}& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,2}}{E}_{\mathrm{2,4}}\oplus E_{\mathrm{1,3}}{E}_{\mathrm{3,4}}\\ E_{\mathrm{2,1}}& 0& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& 0& E_{\mathrm{3,4}}\\ E_{\mathrm{4,2}}{E}_{\mathrm{2,1}}\oplus E_{\mathrm{4,3}}{E}_{\mathrm{3,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& E_{\mathrm{4,2}}{E}_{\mathrm{2,4}}\oplus E_{\mathrm{4,3}}{E}_{\mathrm{3,4}}\end{array}\right) \mathrm{a}\mathrm{n}\mathrm{d} E_{i,k}{E}_{k,j}\le E_{i,j}\left(i\ne j\right).$

Situation Ten: $E_{\mathrm{2,2}}{=E}_{\mathrm{4,4}}=0{,E}_{\mathrm{1,1}}<0,E_{\mathrm{3,3}}<0.$

Similar Situation Six, we have,

$E=\left(\begin{array}{cccc}{E}_{\mathrm{1,2}}{E}_{\mathrm{2,1}}\oplus E_{\mathrm{1,4}}{E}_{\mathrm{4,1}}& E_{\mathrm{1,2}}& E_{\mathrm{1,2}}{E}_{\mathrm{2,3}}\oplus E_{\mathrm{1,4}}{E}_{\mathrm{4,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& 0& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,2}}{E}_{\mathrm{2,1}}\oplus E_{\mathrm{3,4}}{E}_{\mathrm{4,1}}& E_{\mathrm{3,2}}& E_{\mathrm{3,2}}{E}_{\mathrm{2,3}}\oplus E_{\mathrm{3,4}}{E}_{\mathrm{4,3}}& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& 0\end{array}\right) \mathrm{a}\mathrm{n}\mathrm{d} E_{i,k}{E}_{k,j}\le E_{i,j}\left(i\ne j\right).$

Situation Eleven: $E_{\mathrm{3,3}}{=E}_{\mathrm{4,4}}=0{, E}_{\mathrm{1,1}}<0{,E}_{\mathrm{2,2}}<0.$

Similar Situation Six, we have,

$E=\left(\begin{array}{cccc}{E}_{\mathrm{1,3}}{E}_{\mathrm{3,1}}\oplus E_{\mathrm{1,4}}{E}_{\mathrm{4,1}}& E_{\mathrm{1,3}}{E}_{\mathrm{3,2}}\oplus E_{\mathrm{1,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,3}}{E}_{\mathrm{3,1}}\oplus E_{\mathrm{2,4}}{E}_{\mathrm{4,1}}& E_{\mathrm{2,3}}{E}_{\mathrm{3,2}}\oplus E_{\mathrm{2,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& 0& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& 0\end{array}\right) \mathrm{a}\mathrm{n}\mathrm{d}{E}_{i,k}{E}_{k,j}\le E_{i,j}\left(i\ne j\right).$

Situation Twelve: $E_{\mathrm{1,1}}=E_{\mathrm{2,2}}=E_{\mathrm{3,3}}=0,E_{\mathrm{4,4}}<0.$

By equation (21) we obtain $E_{\mathrm{4,4}}=\mathit{max}\left\{E_{\mathrm{4,1}}+E_{\mathrm{1,4}},E_{\mathrm{4,2}}+E_{\mathrm{2,4}},E_{\mathrm{4,3}}+E_{\mathrm{3,4}}\right\}$ . So, we have,

$E=\left(\begin{array}{cccc}0& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& 0& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& 0& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& E_{\mathrm{4,1}}{E}_{\mathrm{1,4}}\oplus E_{\mathrm{4,2}}{E}_{\mathrm{2,4}}\oplus E_{\mathrm{4,3}}{E}_{\mathrm{3,4}}\end{array}\right)\mathrm{a}\mathrm{n}\mathrm{d} E_{i,k}{E}_{k,j}\le E_{i,j}\left(i\ne j\right).$

Situation Thirteen: $E_{\mathrm{1,1}}=E_{\mathrm{2,2}}=E_{\mathrm{4,4}}=0,E_{\mathrm{3,3}}< 0 .$

By equation (16) we know $E_{\mathrm{3,3}}=\mathit{max}\left\{E_{\mathrm{3,1}}+E_{\mathrm{1,3}},E_{\mathrm{3,2}}+E_{\mathrm{2,3}},E_{\mathrm{3,4}}+E_{\mathrm{4,3}}\right\}$ . So, we have,

$E=\left(\begin{array}{cccc}0& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& 0& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& E_{\mathrm{3,1}}{E}_{\mathrm{1,3}}\oplus E_{\mathrm{3,2}}{E}_{\mathrm{2,3}}\oplus E_{\mathrm{3,4}}{E}_{\mathrm{4,3}}& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& 0\end{array}\right)\mathrm{a}\mathrm{n}\mathrm{d} E_{i,k}{E}_{k,j}\le E_{i,j}\left(i\ne j\right).$

Situation Fourteen: $E_{\mathrm{1,1}}=E_{\mathrm{3,3}}=E_{\mathrm{4,4}}=0,E_{\mathrm{2,2}}< 0.$

By equation (11) we get $E_{\mathrm{2,2}}=\mathit{max}\left\{E_{\mathrm{2,1}}+E_{\mathrm{1,2}},E_{\mathrm{2,3}}+E_{\mathrm{3,2}},E_{\mathrm{2,4}}+E_{\mathrm{4,2}}\right\}$ . So, we have,

$E=\left(\begin{array}{cccc}0& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& E_{\mathrm{2,1}}{E}_{\mathrm{1,2}}\oplus E_{\mathrm{2,3}}{E}_{\mathrm{3,2}}\oplus E_{\mathrm{2,4}}{E}_{\mathrm{4,2}}& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& 0& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& 0\end{array}\right)\mathrm{a}\mathrm{n}\mathrm{d}{E}_{i,k}{E}_{k,j}\le E_{i,j}\left(i\ne j\right).$

Situation Fifteen: $E_{\mathrm{2,2}}=E_{\mathrm{4,4}}=E_{\mathrm{3,3}}=0,E_{\mathrm{1,1}}<0.$

By equation (6) we get $E_{\mathrm{1,1}}=\mathit{max}\left\{E_{\mathrm{1,2}}+E_{\mathrm{2,1}},E_{\mathrm{1,3}}+E_{\mathrm{3,1}},E_{\mathrm{1,4}}+E_{\mathrm{4,1}}\right\}$ . So, we have,

$E=\left(\begin{array}{cccc}{E}_{\mathrm{1,2}}{E}_{\mathrm{2,1}}\oplus E_{\mathrm{1,3}}{E}_{\mathrm{3,1}}\oplus E_{\mathrm{1,4}}{E}_{\mathrm{4,1}}& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& 0& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& 0& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& 0\end{array}\right)\mathrm{a}\mathrm{n}\mathrm{d} E_{i,k}{E}_{k,j}\le E_{i,j}\left(i\ne j\right).$

Situation Sixteen: $E_{\mathrm{1,1}}=E_{\mathrm{2,2}}=E_{\mathrm{4,4}}=E_{\mathrm{3,3}}=0.$ So, we have,

$E=\left(\begin{array}{cccc}0& E_{\mathrm{1,2}}& E_{\mathrm{1,3}}& E_{\mathrm{1,4}}\\ E_{\mathrm{2,1}}& 0& E_{\mathrm{2,3}}& E_{\mathrm{2,4}}\\ E_{\mathrm{3,1}}& E_{\mathrm{3,2}}& 0& E_{\mathrm{3,4}}\\ E_{\mathrm{4,1}}& E_{\mathrm{4,2}}& E_{\mathrm{4,3}}& 0\end{array}\right)\mathrm{a}\mathrm{n}\mathrm{d} E_{i,k}{E}_{k,j}\le E_{i,j}\left(i\ne j\right).$

Applications

The rigorous mathematical proof in the above section laid the mathematics foundation for the effective multiple-noise removal. The idempotent of the multiplicative semigroup of higher tropical matrices has been a hot topic in mathematics, and the multiple-noise removal is to harness the tropical algebra’s basic properties. Due to its inherent filtering property, the tropical algebra will open up a totally new window for noise removal, some advanced applications include 3-D printing technology^41–43 and machine learning. ⁴⁴

Tropical addition can replace traditional addition as a more effective filter in denoising. We consider that

$\left(\stackrel{n}{\underset{i=1}{\oplus }}{\mathrm{\epsilon }}_i\oplus \mathrm{\phi }\right)\oplus \mathrm{\phi }=\mathrm{\phi }\oplus \mathrm{\phi }=\mathrm{\phi }$

(22)

where ${\mathrm{\epsilon }}_i\left(1\le i\le n\right)$ is the noise, $\mathrm{\phi }$ is a variable, $\left|{\mathrm{\epsilon }}_i\right|\le \left|\mathrm{\phi }\right|$ , and it can be the amplitude, velocity, or others. As an example,

$\left(2.3\oplus 4.7\oplus 1.9\oplus 5\right)\oplus 5=5$

where ${\epsilon }_1=2.3,{\epsilon }_2=4.7,{\epsilon }_3=1.9,\phi =5$ .

The main method used for mean filtering is the neighborhood average method. ⁴⁵ The basic principle of mean filtering is to replace the pixel values in the original image with mean. That is, for the original pixel $f\left(x,y\right)$ , select a template $h\left(x,y\right)$ , which consists of several pixels in its nearest neighbor, find the mean value of all pixels in the template, and then assign the mean to the current pixel $f\left(x,y\right)$ as the grayscale $g\left(x,y\right)$ on the processed image at that point. Tropical algebra can be applied to the mean filter of multiple-noise. For example,

Traditional algebra algorithm

$\begin{array}{l}\left(\begin{array}{ccc}1& 7& 9\\ 4& 10& 8\\ 2& 5& 6\end{array}\right)\left(\begin{array}{ccc}\frac{1}{9}& \frac{1}{9}& \frac{1}{9}\\ \frac{1}{9}& \frac{1}{9}& \frac{1}{9}\\ \frac{1}{9}& \frac{1}{9}& \frac{1}{9}\end{array}\right)=\left(\begin{array}{ccc}\frac{51}{9}& \frac{45}{9}& \frac{43}{9}\\ \frac{48}{9}& \frac{42}{9}& \frac{44}{9}\\ \frac{50}{9}& \frac{47}{9}& \frac{46}{9}\end{array}\right)\\ f\left(x,y\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}h\left(x,y\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}g\left(x,y\right)\end{array}$

where

$\begin{array}{l}g\left(\mathrm{1,1}\right)=\left(7\times \frac{1}{9}\right)+\left(9\times \frac{1}{9}\right)+\left(4\times \frac{1}{9}\right)+\left(10\times \frac{1}{9}\right)+\left(8\times \frac{1}{9}\right)+\left(2\times \frac{1}{9}\right)+\left(5\times \frac{1}{9}\right)+\left(6\times \frac{1}{9}\right)\\ =\frac{7}{9}+1+\frac{4}{9}+\frac{10}{9}+\frac{8}{9}+\frac{2}{9}+\frac{5}{9}+\frac{6}{9}\\ =\frac{51}{9}.\end{array}$

Topical algebra algorithm

$\begin{array}{l}\left(\begin{array}{ccc}1& 7& 9\\ 4& 10& 8\\ 2& 5& 6\end{array}\right)\left(\begin{array}{ccc}\frac{1}{9}& \frac{1}{9}& \frac{1}{9}\\ \frac{1}{9}& \frac{1}{9}& \frac{1}{9}\\ \frac{1}{9}& \frac{1}{9}& \frac{1}{9}\end{array}\right)=\left(\begin{array}{ccc}\frac{91}{9}& \frac{91}{9}& \frac{91}{9}\\ \frac{91}{9}& \frac{82}{9}& \frac{91}{9}\\ \frac{91}{9}& \frac{91}{9}& \frac{91}{9}\end{array}\right)\\ f\left(x,y\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}h\left(x,y\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}g\left(x,y\right)\end{array}$

where

$\begin{array}{l}g\left(\mathrm{1,1}\right)=\left(7\otimes \frac{1}{9}\right)\oplus \left(9\otimes \frac{1}{9}\right)\oplus \left(4\otimes \frac{1}{9}\right)\oplus \left(10\otimes \frac{1}{9}\right)\oplus \left(8\otimes \frac{1}{9}\right)\oplus \left(2\otimes \frac{1}{9}\right)\oplus \left(5\otimes \frac{1}{9}\right)\oplus \left(6\otimes \frac{1}{9}\right)\\ =\frac{64}{9}\oplus \frac{82}{9}\oplus \frac{37}{9}\oplus \frac{91}{9}\oplus \frac{73}{9}\oplus \frac{19}{9}\oplus \frac{46}{9}\oplus \frac{55}{9}\\ =\mathit{max}\left\{\frac{64}{9},\frac{82}{9},\frac{37}{9},\frac{91}{9},\frac{73}{9},\frac{19}{9},\frac{46}{9},\frac{55}{9}\right\}\end{array}$

$=\frac{91}{9}$

In the above example, the calculation process and results of traditional algebraic algorithm and tropical algebraic algorithm are shown. It can be calculated that the variance S₁ = 0.10 of the nine gray values of $g\left(x,y\right)$ obtained by traditional algebra, the variance S₂ = 0.09 of the nine gray values of $g\left(x,y\right)$ obtained by tropical algebra. Obviously, S₁ is greater than S₂, so the $g\left(x,y\right)$ obtained by the tropical algebra method fluctuates less and more stable than transmission. Therefore, the value obtained by the filter used by the tropical algebra is more ideal.

Tropical addition can replace traditional addition as a more effective filter in denoising. In 2020, Nahar ⁴⁶ proposed a new method combined with hybrid soft computing technology-adaptive wavelet transform (ASWT) filtering. It can be concluded that tropical addition is the optimal scheme for noise filtering, and the filter based on tropical addition can achieve the filtering effect more efficiently and easily.

Now consider an example of optimization and constraints based on tropical algebra,

${\left(x-10\right)}^2+y^2\to \mathit{min}.$

(23)

$s.t.y\left(x\right)=3x\oplus \left(x\otimes 4\right)$

(24)

The constraints of tropical polynomials are shown in equation (24). It is obvious from Figure 1 that the optimal value is 10.

Figure 1.

Optimization under the constraint of the tropical polynomial $y\left(x\right)=3x\oplus \left(x\otimes 4\right)$ .

Discussion and conclusion

Tropical algebra has broad application prospects in data mining. For example, tropical algebra can filter out noise very efficiently, and the filter based on tropical algebra has obvious advantages over the filter based on traditional algebra.^{47, 48} In addition, tropical algebra can also be applied to optimal control, for example, tropical algebra can be introduced into Nadeem-Faisal-He algorithm ⁴⁹ and Chun-Hui He algorithm. ⁵⁰ With the continuous in-depth study of tropical algebra, many scholars have excavated more tropical algebra methods to solve practical problems, such as variational iteration method, ⁵¹ homotopy perturbation method,^52-54 and Rabbani-He-Duz method ⁵⁵ . In the variational principle, the functional stationary point can be obtained by using tropical algebra instead of the traditional algorithm so as to solve practical problems more effectively. ⁵⁶

Tropical algebra has laid the foundations for an abundance of noise removal in various fields, for example, imagine processing and machine learning, and it is a captivating example of the connection between mathematics and practical applications, demonstrating the importance of idempotents in tropical algebra.