A Unified Framework for Generalized Arithmetic: Hyperoperations, Tropical Geometry, and Statistical Moments

A Unified Framework for Generalized Arithmetic: Commutative Operations, Tropical Geometry, and Statistical Moments Mohammed Farhaan, Sanchit Kamat November 1, 2025 Abstract In this paper, we investigate the role of commutative Bennett operations (commutative hyperoperations) in Min-Plus and Max-Plus algebras, and in the dequantization of classical polynomials into tropical polynomials. We reformulate the dequantization process using commutative Bennett operations, yielding a framework that suggests potential alternatives to Maslov dequantization. The formal development of these alternatives is left as an open problem. We apply these commutative Bennett operations to derive generalized forms of the raw and central moments, emphasizing the value of studying these moments collectively rather than in isolation. This motivates the introduction of structural constraints on the family of resulting coefficients, along with interpretations for such structure. The paper further proposes an iterative method for fitting probability density functions to empirical data, including the use of a simple kernel K(x)=x. Additional statistical constructions-such as generalized inner products, covariance, and Pearson’s correlation coefficient-are reexpressed through commutative Bennett operations. Finally, we introduce two approaches for defining commutative Bennett operations between matrices: one by equipping each matrix with a compatible algebra and transferring the operations to these derived structures, and another derived directly from our generalized inner product. 1 Introduction and Foundational Concepts The extension of binary arithmetic operations beyond the foundational four—addition, multiplication, exponentiation, and tetration—has long been of interest to mathematicians seeking deeper algebraic and analytic structure. Early attempts to formalize these higher operations can be traced to the works of Albert A. Bennett, who in his 1914 paper A General Form for the Fundamental Operations of Arithmetic [1] proposed one of the earliest unified formulations of recursively defined arithmetic operations. Subsequent developments by Löwner, Goodstein, and later Knuth broadened this viewpoint into what is now recognized as the “Bennett sequence’’ or 1 the extended hierarchy of commutative Bennett operations. These operations generalize classical arithmetic through recursive patterns that preserve structural consistency across the entire hierarchy. Beyond their aesthetic appeal, commutative Bennett operations have emerged as useful tools in diverse mathematical fields. Their recursive nature provides a mechanism for uniformly expressing algebraic transformations, functional iteration, and scale transitions. In recent decades, commutative variants of these operations have gained particular interest for applications in idempotent analysis, non-linear algebra, and tropical mathematics. 1.1 The Bennett Sequence and Its Mathematical Significance Immediately following Bennett’s original formulation, researchers began to study the properties and interactions of the resulting operational hierarchy. The Bennett sequence highlights an elegant consistency: each operation serves as a natural extension of the previous one, governed by a recursion that mirrors the jump between addition and multiplication, or between multiplication and exponentiation. This recursive viewpoint offers a principled route to generalization, enabling forms of arithmetic that interpolate between or transcend classical operations. Modern treatments often reinterpret the Bennett sequence through functional iteration, ordinal analysis, and algebraic operads. These viewpoints reveal deep symmetries not visible at the level of elementary arithmetic and position commutative Bennett operations as a natural language for expressing generalized algebraic phenomena. 1.2 Commutative Bennett Operations in Tropical Mathematics A second major theme relevant to this work is the development of Tropical Geometry, a field that emerged from the study of idempotent semirings and optimization theory. The tropical semiring—built on the operators min (or max) and +—first appeared in early work on discrete event systems and convex geometry, and was later formalized in the context of algebraic geometry 2 in the 1990s and early 2000s by Speyer, Sturmfels, Litvinov, Maslov, and others. Central to tropical mathematics is the process of dequantization, sometimes referred to as Maslov Dequantization, which transforms classical polynomials into piecewise-linear tropical polynomials via logarithmic scaling limits. Traditionally, this dequantization is expressed analytically through asymptotic transformations of sums and products. In this paper, we reformulate the dequantization process using commutative Bennett operations. This not only provides algebraic clarity but opens the possibility of constructing alternative dequantizations by modifying the Bennett–type recursion underlying the operation hierarchy. Such alternatives may yield new tropical-like structures, a problem we identify as open for future investigation. 1.3 Structure and Contributions of this Paper This paper is structured to demonstrate the utility of commutative Bennett operations in both tropical mathematics and statistics. We begin by formally establishing the notation and algebraic properties of the commutative Bennett hierarchy and its inverses (Section 2). We then reinterpret tropical dequantization through this framework, highlighting how Bennett operations naturally arise in the Min–Plus and Max–Plus settings (Section 4). Following this, we introduce a generalized theory of statistical moments derived entirely through commutative Bennett operations. We argue that raw moments, central moments, and related coefficients such as skewness and kurtosis should be understood as a collective family rather than isolated quantities (Sections 5 and 6). This motivates imposing structural constraints on these families of coefficients and interpreting them algebraically. Finally, we present two approaches to defining commutative Bennett operations between matrices: (1) attaching a compatible algebra to each matrix and transferring operations to these induced structures, and (2) deriving the matrix operations directly from a generalized inner product. These constructions allow us to reinterpret covariance and Pearson’s correlation coefficient in Bennett-operational terms, further illustrating the broad applicability of the framework. 3 1.4 Summary and Application of Commutative Bennett Operations This paper is structured to demonstrate the utility of Commutative Bennett Operations through a comprehensive Application of this generalized arithmetic framework across two distinct mathematical domains: tropical mathematics and statistical analysis. 1.4.1 Summary of Contributions We begin by formally establishing the notation and the algebraic properties of the commutative Bennett hierarchy and its inverses (Section 2). The core contributions of this work are presented as follows: 1. Application to Tropical Geometry (Section 4): We reinterpret the process of tropical dequantization through the Bennett operational framework. This provides a novel, purely algebraic foundation for understanding the transition from classical to tropical arithmetic, highlighting how Bennett operations naturally arise in the Min–Plus and Max–Plus settings. 2. Application to Statistical Analysis (Sections 5 and 6): We introduce a generalized theory of statistical moments derived entirely through Commutative Bennett Operations. We argue that raw moments, central moments, and related coefficients such as skewness and kurtosis should be understood as a collective family rather than isolated quantities. This operational interpretation motivates imposing structural constraints on these families of coefficients. 3. Application to Matrix Algebra and Correlation (Section 7): We define commutative Bennett operations between matrices using two approaches: by attaching a compatible algebra to each matrix, and by deriving the operations from a generalized inner product. These constructions allow us to reinterpret classical concepts like covariance and Pearson’s correlation coefficient in Bennett-operational terms, further illustrating the broad applicability of the framework. 4 2 The Commutative operation Hierarchy and Notation The hyperoperation hierarchy is built upon the principle that each operation in the sequence is a repeated iteration of the previous one. Following this principle, we define ⊙m xas the commutative hyperoperation of order mwith base x, and Θm xas its corresponding inverse operation. Let ⊙m xbe the commutative hyperoperation and Θm xbe the inverse of ⊙m x. 2.1 Hyper Summation Notation We introduce the hyper summation notation as a concise way to represent the repeated application of the hyperoperation ⊙m xover a sequence of terms ai. n      m · x      i=0 ai=ae⊙m xa1⊙m xa2⊙m x· · · ⊙m xan This notation generalizes the classical summation (∑) and product (∏) operators, which correspond to specific, low orders of the hyperoperation m. The operator design itself visually conveys three key parameters: the operational order m, the type of operation (represented by the dot ·), and the base of the operation x. Similarly, the inverse operation Θm xcan be applied repeatedly over a sequence of terms bi. The notation for a generalized sum of nterms biusing the inverse hyperoperation is: n      m x      i=0 bi=b0Θm xb1Θm x· · · Θm xbn 5 This notation, distinguished by the bar, allows for a comprehensive algebraic treatment of both the primary and inverse hyperoperations. 2.2 Iterated Functions and the Recursive Definition In the context of the operational hierarchy, it becomes useful to define iterated applications of the logarithmic and exponential functions. We also denote the repeated logarithm and exponential functions: log(log(· · · xtimes · · · log(z))) = logm x(z) exp(exp(· · · xtimes · · · exp(z))) = expm x(z) These iterated functions are necessary components for the recursive definition of the hyperoperation itself, facilitating movement between operational levels. The core of the hyperoperation framework is the generalized recursive structure, which defines the order mhyperoperation in terms of the order m−1hyperoperation. This relationship, often known as the defining identity, links addition, multiplication, and exponentiation in a uniform manner. And the operation: a⊙m xb=exp (log(a)⊙m x−1log(b))(1) a⊙m xb=exp x(log x(a) + log x(b)) (1.2) This fundamental recursive identity allows for the systematic construction of the entire hierarchy of operations. 6 3 Properties of the Commutative The generalized operations maintain several structural properties inherent to the lower-level arithmetic operations, provided they are appropriately defined within the commutative context. For the hyperoperation ⊙m x, we have the following properties: 1. Commutativity: a⊙m xb=b⊙m xa This property is crucial for the simplification and algebraic manipulation of the hypersummations and is generally preserved for all orders m≥0. 2. Recursive Definition for m∈Z+: a⊙m xb=exp (log(a)⊙m x−1log(b))(1) This is the same recursive identity shown in Equation 1 3.1 Negative Ordered Commutative Operations To ensure the system is complete and closed, the inverse Commutativ Operation are defined for negative operational orders. This extends the scope of the framework to include both the forward operations and their inverses. For m < 0, we define the inverse operation ⊙−m x: a⊙m −xb=log x(exp x(a) + exp x(b)) = log(exp(a)⊙m −x+1 exp(b)) This definition provides a smooth transition across the zero order of the hyperoperation, ensuring algebraic consistency. 7 Throughout this paper, we denote logm(a) = log(a)and ma=expm(a) = exp(a), serving as simplifying aliases for the generalized exponential and logarithmic functions within this specific operational context. 4 Connection to Tropical Geometry and Maslov Dequantization The hyperoperation framework finds a powerful, non-trivial connection to the field of Tropical Geometry. Tropical Geometry, often described as an algebraic geometry in the Max-Plus algebra, offers a method of ”dequantizing” classical algebraic varieties to produce piecewise-linear structures that are simpler to analyze. This connection is fundamental to the work presented here. Max Plus Algebra is defined as the semiring (R,⊕,⊗), where ⊕denotes tropical addition and ⊗denotes tropical multiplication. This semiring replaces the standard operations of addition and multiplication with maximization and standard addition, respectively. • Tropical addition: ⊕∼ =max() • Tropical multiplication: ⊗∼ =+ • Identities: id⊕=−∞ and id⊗= 0 The link between classical algebra and tropical algebra is formally established through the Tropical functor Trop: We use a Tropical functor Trop that takes us from the category of Polynomials to the category of Tropical polynomials: (R, +,×)Trop() −−−−→ (R,⊕,⊗) This functor acts as a conceptual bridge between the two categories. 8 βp (3−2) ={. . . , β−4 (3−2), β−3 (3−2), β−2 (3−2), β−1 (3−2), β0 (3−2), β1 (3−2), β2 (3−2), β3 (3−2), β4 (3−2), . . .} In general, we can expand our set to include our hyper for a fixed difference (r1−r2): β(p,k) (r1−r2)=βp (r1−r2) βp (r1−r2)={. . . , β−2 (r1−r2), β−1 (r1−r2), β0 (r1−r2), β1 (r1−r2), β2 (r1−r2), . . .} And so the hyper general coefficients of skewness are defined as the Cartesian product of these two infinite sets, where the condition r1−r2= 1 is imposed: βp 1={(a, b)|a=β(r1−r2)∈β1, b ∈βp (r1−r2),∀r1−r2= 1 =r1, r2∈N} what we get is a 2D lattice that encodes the nonlinearity of my dataset and what this allows us to do is a path to convert statements about our dataset into statements about this lattice and the values obeserved at each point of the lattice 6.2 Hyper Coefficients of Skewness For the given data, we assume that µ(r, µ′ r0, P, k, m) = µ(r′−r, µ′ r0r0, P, k, m) and we can safely assume that β((−3)−(−4)) =β(4−3) 15 We must construct our βset as the following: β1={. . . , β((−3)−(−4)), β((−2)−(−3)), β((−1)−(−2)), β(0−(−1)), β(1−0), β(2−1), β(3−2), β(4−3), . . . } We expect β1to be an infinite set of real numbers, and we are interested in the structure of our βset. We also construct what we define as the hyper coefficients of skewness as: βp (3−2) ={. . . , β−4 (3−2), β−3 (3−2), β−2 (3−2), β−1 (3−2), . . . , β3 (3−2), β4 (3−2), . . . } In general: βp (r1−r2)={. . . , β−2 (r1−r2), β−1 (r1−r2), β0 (r1−r2), β1 (r1−r2), β2 (r1−r2), . . . } And so the hyper general coefficients of skewness can be defined as βp 1={(a, b)|a=β(r1−r2)∈β1, b ∈βp (r1−r2)} with the condition that ∀r1−r2= 1 |r1, r2∈N Using the set βρ 1, we want to derive some metric that can give a more natural interpretation of all the moments in βρ 1. For this, we must answer the question: ”What is a useful way to study this set of real numbers either as a sequence of an unordered set of reals to obtain a measure on skewness?” 6.3 Metric for Interpretation With this geometric interpretation of our dataset, we are hunting for a natural interpretation of this lattice and its properties .Using the set βp 1, we want to exploit its inherent Z×Zstructure 16 as we get a metric which corresponds to the lattice We cannot interpret this as a distribution over the 2D lattice as the signage of each beta values allows us to determine important properties about the data set. The study of this data set could be the subject of future study. 6.4 Kernel Density Functions and Bandwidth Selection 6.4.1 Kernel Density Estimation (KDE) Kernel density functions are used to fit a given dataset to a given kernel K(x), which is an arbitrary probability distribution function. For a dataset {xi}n i=1 and a kernel K(x)that satisfies the following: 1. K(x)≥0,∀x 2. ∫∞ −∞ K(x)dx = 1 3. K(u) = K(−u)(optional for a symmetric distribution) 4. ∫∞ −∞ uK(u)du = 0 (mean is centered at origin) 5. ∫∞ −∞ u2K(u)du < ∞(variance is finite) We propose the following generalization: ˆ f0 n(x) = 1 nh n ∑ i=0 K[x−xi h]where h > 0 Here, nis the sample size and his the bandwidth. 17 6.4.2 Iterative Bandwidth Selection Define h0and h1such that h0< h1. The bandwidth h0causes overfitting, and h1causes oversmoothing. We define the initial error metric ˆ h0 01(x)as the mean square error (MSE) between the two density estimates: ˆ h0 01(x) = (ˆ f0 n0(x)−ˆ f0 n1(x))2 where ˆ f0 n0(x) = 1 nh0 n ∑ i=0 K[x−xi h0] ˆ f0 n1(x) = 1 nh1 n ∑ i=0 K[x−xi h1] We note that ˆ h0 01(x)≥0, as it is a Mean Square Error. The derivative is given by (using dK du for the kernel derivative): dˆ h0 01(x) dx = 2 (1 nh0 n ∑ i=0 K[x−xi h0]−1 nh1 n ∑ i=0 K[x−xi h1])( 1 nh2 0 n ∑ i=0 dK du [x−xi h0]−1 nh2 1 n ∑ i=0 dK du [x−xi h1]) Since we may still have some over/underfitting, we propose an iterative process that allows us to converge into the probability distribution of any given dataset.(that is if we set K(x) = x) If we assume ˆ h0 01(x)<ˆ h0 12(x), such that ˆ h0 01 is overfitting and ˆ h0 12 is oversmoothing, we define the next order MSE: ˆ h1 01 =(ˆ f1 0(x)−ˆ f1 1(x))2 2 18 where the first-order densities are: ˆ f1 0(x) = 1 nˆ h01 n ∑ i=0 K[x−xi ˆ h0 01 ] ˆ f1 1(x) = 1 nˆ h12 n ∑ i=0 K[x−xi ˆ h0 12 ] We can recursively define this iteratively on the condition that ˆ hj 01 is overfitting and ˆ hj 12 is oversmoothing. ˆ fj+1 0(x) = 1 nˆ hj 01 n ∑ i=0 K[x−xi ˆ hj 01 ] ˆ fj+1 1(x) = 1 nˆ hj 12 n ∑ i=0 K[x−xi ˆ hj 12 ] ˆ hj+1 01 =(ˆ fj+1 0−ˆ fj+1 1)2 2 6.4.3 Commutative General Kernel Plots We generalize ˆ hj 01 as: ˆ H01(j, 0) = ˆ hj 01 ˆ Hn1(j, 1) =       ˆ fj n(x) ˆ fj n(x) ˆ fj n(x)       Θj 42Θj 32 19 ˆ H01(j, k) = [[ˆ fj 0Θe (k+n)ˆ fj 1]Θj (k+n)]Θj (k+n)−2 These are called the kth order commutative Errors. Based on this definition, the commutative general kernel plots are: ˆ Fj+1 0(x) = 1 nˆ H0(j, k) n ∑ i=0 K[x−xi ˆ H0(j, k)] ˆ Fj+1 1(x) = 1 nˆ H1(j, k) n ∑ i=0 K[x−xi ˆ H1(j, k)] 6.4.4 Analysis with K(x) = x We were inspired by the Lebesgue Measure and wanted to test our density plots by using simply f(x) = x? We set K(x) = x. Calculating ˆ f1 n(x)with K(x) = x: ˆ f0 n0(x) = 1 nh0 n ∑ i=0 x−xi h0 =1 nh2 0 n ∑ i=0 (x−xi) ˆ f0 n1(x) = 1 nh2 1 n ∑ i=0 (x−xi) ˆ h0 01(x) = 1 2(1 nh2 0 n ∑ i=0 (x−xi)−1 nh2 1 n ∑ i=0 (x−xi))2 Let L=∑n i=0(x−xi). 20 ˆ h0 01(x) = 1 2[1 n n ∑ i=0 (x−xi)]2(1 h2 0 −1 h2 1)2 =L2(h2 1−h2 0)2 2n2h4 0h4 1 Now ˆ f1 n0(x)is evaluated: ˆ f1 n0(x) = 1 nˆ h0 01 n ∑ i=0 x−xi ˆ h0 01 =1 n(ˆ h0 01)2(n ∑ i=0 (x−xi))=L n(ˆ h0 01)2 Substituting the expression for ˆ h0 01(x): ˆ f1 n0(x) = L n×1 (L2(h2 1−h2 0)2 2n2h4 0h4 1)2=L n×4n4h8 0h8 1 L4(h2 1−h2 0)4 ˆ f1 n0(x) = 4n3h8 0h8 1 L3(h2 1−h2 0)4=4n3h8 0h8 1 (∑n i=0(x−xi))3(h2 1−h2 0)4 If we consider the limit h0→h−and h1→h+(where his a perfect latent bandwidth), the expression becomes: ˆ f1 n0(x) = 4n3(h−)8(h+)8 ((h+)2−(h−)2)4×(n ∑ i=0 (x−xi))−3 Or, in terms of inverse squared bandwidths: ˆ f1 n0(x) = 4n3 (1 h2 0 −1 h2 1)4(∑n i=0(x−xi))3 Perhaps this coefficient is useful. 21 6.5 Commutative Operations on Matrices We consider two approaches,but we only elaborate on the second one. Its likely that they lead to completely different operations due to the fact that we cant : 1. Evaluate Commutative Operations for existing matrix representations of algebras, like complex numbers and other Clifford Algebras. 2. Generalize the Euclidean inner product. 6.5.1 Constructor Reduction (Approach 1) Given a Cl(p,q,r)construction and a matrix M, what we want is (M(Cl(p, q, r)), which is a Matrix representation that takes the Clifford Algebraand reduces it to a matrix M: Matrix Mrepresentation →Cl(pM, qM, rM) We want to perform our hyper operations within Cl(pM, qM, rM)instead of directly acting on the matrices. 6.5.2 Generalizing Euclidean Inner Product (Approach 2) Let Aand Bbe n×nmatrices, xbe an n-vector (row), and ybe an n-vector (column). Hyper Inner Products The standard inner product is: ⟨x, y⟩0=⟨x, y⟩= n ∑ i=1 (xiyi) 22 The generalized m-hyper inner product is: ⟨x, y⟩k=       n           e · 0           i=0 (xi⊙e myi)       Hyper Matrix Operations Any given matrix multiplication is (AB)ij = n ∑ k=1 aikbkj = n ∑ k=1 aik ⊙e 1bkj The m-th hyper operation ⊙e mof Aon Bis defined as: (A⊙e mB)ij =       n           e · m           i=0 aik ⊙e m+1 bkj        Hyper Covariance and Hyper Correlation First, we define the deviation vector σx: σx=          x0−x x1−x . . . xn−1−x          The mean deviation vector σy: σy=          y0−y y1−y . . . yn−1−y          23 The covariance of xand y: cov(x, y) =        n           e · 0           i=0 (xi−x)(yi−y)       Θe 1n Similarly, the hyper covariance or k-covariance is covk(x, y) = ⟨σx, σy⟩kΘe k+1n The corresponding variances are: vark(x) = covk(x, x)vark(y) = covk(y, y) Karl Pearson’s k Coefficient of correlation is given as rk(x, y): rk(x, y) = covk(x, y) √vark(x)vark(y) Karl Pearson’s wildly impractical coefficient of correlation is given as: rk0,12(x, y) = covk0(x, y) √vark1(x)vark2(y) Conjecture: (loosely speaking) k0, k1, k2that maximize or minimize rk0,12(x, y)contain information about the nature of correlation. 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