pith. sign in

arxiv: 2412.07153 · v3 · submitted 2024-12-10 · 🧮 math.RA

t-Product and t-STP of Cubic Matrices With Application to Hyper-Networked Systems

Daizhan Cheng , Zhengping Ji This is my paper

Pith reviewed 2026-05-23 07:50 UTC · model grok-4.3

classification 🧮 math.RA
keywords cubic matricest-productt-STPsemi-tensor producthyper-networked systemsdynamic controlevolutionary gamesupply chain
0
0 comments X

The pith

The t-STP on cubic matrices integrates t-product and dimension-keeping multiplication to couple subsystem dynamics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes a new operation called t-STP for cubic matrices to model dynamic control systems with interacting parts. It addresses the strict size rules of the t-product by adding a dimension-keeping semi-tensor product, then merges the two so subsystems can interact across frontal slices without decoupling. The resulting algebra supports standard structures like groups and rings. The method is used to build models for hyper-networked evolutionary games that describe supply chain interactions.

Core claim

The central discovery is that the t-semi-tensor product (t-STP) of cubic matrices, formed by combining the t-product with the dimension-keeping semi-tensor product, permits coupled dynamics between subsystems while allowing arbitrary dimensions, and this operation generates algebraic structures such as groups, rings, modules, and Lie groups that support applications to hyper-networked systems.

What carries the argument

t-STP, the integration of the t-product and DK-STP on cubic matrices that enables coupled frontal slice interactions with flexible dimensions.

If this is right

  • Cubic matrix based linear and nonlinear controlled dynamics can be constructed with preserved interactions.
  • Algebraic structures like groups, rings, modules and Lie groups arise naturally from the t-STP.
  • The framework applies directly to modeling supply chain interactions via hyper-networked evolutionary games.
  • Greater flexibility is achieved in representing coupled subsystems compared to prior products.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the t-STP holds its properties, it could be extended to simulate real-time control in multi-agent systems.
  • Neighbouring problems in tensor algebra might benefit from similar dimension-keeping integrations.
  • A concrete test would involve checking closure on random cubic matrices of mismatched sizes.

Load-bearing premise

The newly defined t-STP preserves the algebraic closure and interaction properties required for the hyper-networked game model to stay consistent.

What would settle it

A calculation showing that t-STP of two specific cubic matrices produces a result that violates closure or loses the cross-slice couplings would falsify the claim.

Figures

Figures reproduced from arXiv: 2412.07153 by Daizhan Cheng, Zhengping Ji.

Figure 1
Figure 1. Figure 1: Cubic Matrices • ⋉ : dimension-keeping semi-tensor product. • ⋆: t-product of cubic matrices. • ⋉ ∗: t-STP. • Γ: universal homomorphism. • gl∗(m × n × s, R): t-STP based general linear algebra. • GL∗(m × n × s, R): t-STP based general linear group. • ≃: homomorphism. • ∼=: isomorphism. II. PRELIMINARIES A. t-Product of Cubic Matrices The data of an order 3 hyper-matrix can be arranged into a cub [PITH_FUL… view at source ↗
Figure 2
Figure 2. Figure 2: A Hypergraph (i) GL(m × n, R) := {Im×n + A0 | A0 ∈ Mm×n, and Im×n + A0 is invertible.}. (13) Then GL(m × n, R) is a Lie group, called the general Linear group of dimension m × n. (ii) The Lie algebra of GL(m × n, R) is gl(m × n, R). C. Hypergraph Definition 2.12: [4] A hypergraph H = (V, E) is a couple, where the finite set V = {x1, x2, · · · , xn} is the set of vertices; E = {E1, E2, · · · , Em} is the se… view at source ↗
Figure 4
Figure 4. Figure 4: A Supply Network H IX. HYPER-NETWORKED EVOLUTIONARY GAME Consider a supply hyper-network Σ, described in [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Dual Hyper-graph H∗ Now assume the strategies of xi , denoted by Sx = {H, L}, where H stands for high products, and L stands for low products; the strategies of yj , denoted by Sy = {H, M, L}, where H (L) means receiving high (low) amount of products and high (low) amount of products to pass to the markets, M stands for mixed situation; the strategies of zk are also Sz = {H, L} with similar meanings. Now t… view at source ↗
read the original abstract

Motivated by the study of dynamic control systems, this paper proposes novel algebraic operations on cubic matrices to construct both linear and nonlinear controlled dynamics. The standard t-product of cubic matrices imposes strict dimensional constraints; to resolve this, we first introduce the dimension-keeping semi-tensor product (DK-STP), which generalizes the matrix product for arbitrary dimensions. However, the DK-STP yields decoupled subsystem dynamics because it fails to capture interactions across subsystems corresponding to frontal slices. To overcome this limitation, we propose the t-semi-tensor product (t-STP), an integration of the t-product and the DK-STP that allows for coupled subsystems and greater modeling flexibility. We systematically study the algebraic structures derived from the t-STP over cubic matrices, including groups, rings, modules, and Lie groups. Finally, we obtain t-STP-based dynamic control systems over cubic matrices and demonstrate the utility of this framework by applying it to a hyper-networked evolutionary game modeling supply chain interactions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes the dimension-keeping semi-tensor product (DK-STP) to relax dimensional constraints of the standard t-product on cubic matrices, then defines the t-semi-tensor product (t-STP) as their integration to enable coupled subsystem dynamics. It examines the resulting algebraic structures (groups, rings, modules, Lie groups) over cubic matrices and applies the t-STP framework to construct dynamic control systems, with a demonstration on hyper-networked evolutionary games modeling supply-chain interactions.

Significance. If the algebraic closure, associativity, and frontal-slice coupling properties hold under t-STP, the framework could supply a systematic way to model interacting subsystems in hyper-networked control problems that standard t-products cannot accommodate. The supply-chain game application supplies a concrete, falsifiable test case for the modeling claim.

major comments (2)
  1. [Algebraic structures section (post-definition of t-STP)] The definition and subsequent proofs that t-STP forms groups, rings, and modules must explicitly verify that the operation inherits associativity and closure from both the t-product and DK-STP; the integration step risks breaking these properties, which would invalidate all later algebraic claims.
  2. [Application to hyper-networked evolutionary game] In the evolutionary-game application, the claim that t-STP encodes cross-slice coupling for consistent hyper-networked dynamics rests on the unverified assumption that frontal-slice interactions remain intact; if the new operation decouples slices, the game model loses the intended interaction structure.
minor comments (2)
  1. Define the precise block-wise or slice-wise construction of the t-STP (including any reshaping or permutation steps) with an explicit formula or diagram before the algebraic theorems.
  2. Add a short table contrasting the dimensional requirements and coupling behavior of t-product, DK-STP, and t-STP.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed reading and constructive feedback on our manuscript. Below we respond point-by-point to the major comments. We will revise the manuscript to strengthen the explicit verification of algebraic properties while maintaining that the core claims hold by construction of the t-STP.

read point-by-point responses

  1. Referee: [Algebraic structures section (post-definition of t-STP)] The definition and subsequent proofs that t-STP forms groups, rings, and modules must explicitly verify that the operation inherits associativity and closure from both the t-product and DK-STP; the integration step risks breaking these properties, which would invalidate all later algebraic claims.

    Authors: The t-STP is defined in Section 3 as the integration of the t-product (for frontal-slice coupling) and DK-STP (for dimensional flexibility) via a block-wise construction that directly inherits the associative and closed operations of its components; the proofs of the group, ring, and module structures in Theorems 4.1–4.5 explicitly invoke this inheritance. Nevertheless, we agree that a dedicated preliminary lemma making the inheritance of associativity and closure fully explicit would improve clarity and address the concern about the integration step. We will insert such a lemma in the revised version. revision: yes

  2. Referee: [Application to hyper-networked evolutionary game] In the evolutionary-game application, the claim that t-STP encodes cross-slice coupling for consistent hyper-networked dynamics rests on the unverified assumption that frontal-slice interactions remain intact; if the new operation decouples slices, the game model loses the intended interaction structure.

    Authors: By definition the t-STP retains the t-product component precisely to keep frontal-slice interactions intact while relaxing dimensional constraints via the DK-STP part; this is stated in the construction preceding the application (Section 5) and is used to derive the coupled dynamics equations for the hyper-networked game. The supply-chain example then demonstrates preservation of these interactions through the resulting payoff and strategy-update rules. We will add one clarifying sentence in the application section reiterating that the coupling is inherited directly from the t-product term. revision: partial

Circularity Check

0 steps flagged

No circularity: new operations defined from first principles with independent algebraic verification

full rationale

The paper's chain consists of (1) identifying a dimensional restriction in the existing t-product, (2) defining DK-STP to remove that restriction, (3) defining t-STP as an explicit integration of t-product and DK-STP to restore cross-slice coupling, and (4) verifying that the resulting structures satisfy group/ring/module/Lie-group axioms by direct (if tedious) checking of the definitions. None of these steps reduces a claimed result to its own input by construction; each new operation is introduced by an explicit formula whose properties are then proved rather than presupposed. No fitted parameters, no 'predictions' of data, and no load-bearing self-citations whose content is merely renamed appear. The application section simply instantiates the already-verified algebraic objects inside a supply-chain model; it does not derive the algebraic closure from the model or vice versa. Consequently the derivation remains self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on the well-definedness of two new operations and on the claim that they enable coupled dynamics; these are introduced by definition rather than derived from prior results.

axioms (1)
  • standard math The t-product on cubic matrices satisfies its standard algebraic properties as previously defined in the literature.
    Invoked when combining t-product with DK-STP to form t-STP.
invented entities (2)
  • t-STP operation no independent evidence
    purpose: To integrate t-product and DK-STP so that frontal slices can interact across subsystems of arbitrary dimensions.
    Newly defined in the paper; no independent evidence outside the definitions is given.
  • DK-STP operation no independent evidence
    purpose: To generalize matrix multiplication to cubic matrices of arbitrary dimensions while keeping dimensions.
    Newly defined in the paper to remove strict dimensional constraints.

pith-pipeline@v0.9.0 · 5701 in / 1239 out tokens · 29833 ms · 2026-05-23T07:50:16.507434+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

  1. [1]

    Ahsan, Monoids characterized by the their quasi-inje ctive S-systems, Semigroup F orum, V ol

    J. Ahsan, Monoids characterized by the their quasi-inje ctive S-systems, Semigroup F orum, V ol. 36, 285-292, 1987

    work page 1987

  2. [2]

    Arznagh, G

    D.A. Arznagh, G. Michailidis, Fast randomized algorith ms for t-product based tensor operations and decompositions with applicati ons to imaging data, SIAM J. Imaging Sci. , V ol. 11. No. 4, 2629-2664, 2018

    work page 2018

  3. [3]

    Batcs, D

    D. Batcs, D. Watts, Parameter transformations for impro ved approximate confidence regions in nonlinear least squares, The Annals of Statistics V ol. 9, 1152-1167, 1981. 19

    work page 1981

  4. [4]

    Berge, Graphs and Hypergraphs , North-Holland Pub

    C. Berge, Graphs and Hypergraphs , North-Holland Pub. Comp., New Y ork, 1973

    work page 1973

  5. [5]

    Boothby, Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd Ed., Elsevier, London, 1986

    W.M. Boothby, Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd Ed., Elsevier, London, 1986

    work page 1986

  6. [6]

    Chang, Y

    S. Chang, Y . Wei, T-product tensor-part ii: Tail bounds f or sums of random t-product tensors, Comput. Appl. Math. , V ol. 41, No. 3: 99, 2022

    work page 2022

  7. [7]

    Chen, Thenor-Based Dynamical Systems- Theory and App lications, Springer, Switzerland, 2024

    C. Chen, Thenor-Based Dynamical Systems- Theory and App lications, Springer, Switzerland, 2024

    work page 2024

  8. [8]

    Cheng, H

    D. Cheng, H. Qi, Z. Li, Analysis and Control of Boolean Networks - A Semi-tensor Product Approach , Springer, London, 2011

    work page 2011

  9. [9]

    Cheng, H

    D. Cheng, H. Qi, Y . Zhao, An Introduction to Semi-tensor Product of Matrices and Its Applications , World Scientific, Singapo, 2012

    work page 2012

  10. [10]

    Cheng, On equivalence of matrices, Asian Journal of Math

    D. Cheng, On equivalence of matrices, Asian Journal of Math. , V ol. 23, No. 2, 257-348, 2019

    work page 2019

  11. [11]

    Cheng, Y

    D. Cheng, Y . Wu, G. Zhao, S. Fu, A comprehensive survey on STP approach to finite games, J. Sys. Sci. Compl. , V ol. 34, No. 5, 1666- 1680, 2021

    work page 2021

  12. [12]

    Cheng, M

    D. Cheng, M. Meng, X. Zhang, Z. Ji, Contracted product of hypermatri- ces via STP of matrices, Control Theory Appl. , V ol. 21, No. 3, 265-280, 2023

    work page 2023

  13. [13]

    Cheng, From semi-tensor product of metrices to non-s quare general linear algebra and general linear group, Commu

    D. Cheng, From semi-tensor product of metrices to non-s quare general linear algebra and general linear group, Commu. Inform. Sys. , V ol. 24, No. 1, 1-60, 2024

    work page 2024

  14. [14]

    Cheng, Cross-dimensional mathematics–a foundatio n for STP/STA, (preprint: http:arxiv.org/abs/2406.12920), 2024

    D. Cheng, Cross-dimensional mathematics–a foundatio n for STP/STA, (preprint: http:arxiv.org/abs/2406.12920), 2024

    work page arXiv 2024

  15. [15]

    DeepSeak-AI, DeepSeek-v3 technical report, arXiv:24 12.19437v1, 2024

    work page 2024

  16. [16]

    Dixon, B

    J.D. Dixon, B. Mortimer, Permutation Group , Springer-V erlag, New Y ork, 1996,

    work page 1996

  17. [17]

    Hall, Lie Group, Lie Algebra, and Representations - An Elementary Introduction, Springer, New Y ork, 2003

    B.C. Hall, Lie Group, Lie Algebra, and Representations - An Elementary Introduction, Springer, New Y ork, 2003

    work page 2003

  18. [18]

    Hoover, K

    R.C. Hoover, K. Caudle, K. Braman, A new approach to mult ilinear dynamical systems and control, arXiv:2108.13583, 2021

    work page arXiv 2021

  19. [19]

    Hungerford, Algebra, Springer-V erlag, New Y ork, 1974

    T.W. Hungerford, Algebra, Springer-V erlag, New Y ork, 1974

    work page 1974

  20. [20]

    Isidori, Nonlinear Control Systems , 3rd Ed., Springer-V erlag, New Y ork, 1995

    A. Isidori, Nonlinear Control Systems , 3rd Ed., Springer-V erlag, New Y ork, 1995

    work page 1995

  21. [21]

    Jiang, C

    N. Jiang, C. Huang, Y . Chen, J. Kurths, Bisimulation-ba sed stabilization of probabilistic Boolean control networks with state feedb ack control, Front Inform. Technol. Electron. Eng. , V ol. 21, No. 2, 268-280, 2020

    work page 2020

  22. [22]

    Jost, Dynamic Systems , Springer-V erlag, Berlin, 2015

    H. Jost, Dynamic Systems , Springer-V erlag, Berlin, 2015

    work page 2015

  23. [23]

    Khoromskij, Tensor numerical methods for multidi mensional PDES: Theoretical analysis and initial applications, ESAIM Proc Surv

    B.N. Khoromskij, Tensor numerical methods for multidi mensional PDES: Theoretical analysis and initial applications, ESAIM Proc Surv. , V ol. 48, 1-28, 2015

    work page 2015

  24. [24]

    Kilmer, C.D

    M.E. Kilmer, C.D. Martin, Factorization strategies fo r third-rder tensors, Linear Algebra & Its Appl. , V ol. 435, No. 3, 641-658, 2011

    work page 2011

  25. [25]

    Kilmer, K

    M.E. Kilmer, K. Braman, M. Nao, R.C. Hoover, Third-orde r tensors as operators on matrices: A theoretical and computational f romework with applications in imaging, SIAM J. Mathrix Anal. Appl. , V ol. 34, No. 148-172, 2013

    work page 2013

  26. [26]

    H. Li, G. Zhao, M. Meng, J. Feng, A survey on applications of semi- tensor product method in engineering, Science China, V ol. 61, 010202:1- 010202:17, 2018

    work page 2018

  27. [27]

    R. Li, T. Chu, X. Wang, Bisimulations of Boolean control networks, SIAM J. Opt. , V ol. 51, No. 1, 388-416, 2018

    work page 2018

  28. [28]

    Lim, Tensors and Hypermatrices, in L

    L. Lim, Tensors and Hypermatrices, in L. Hogben (Ed.) Hand- book of Linear Algebra (2nd ed.), Chapter 15, Chapman and Hall/CRC.https://doi.org/10.1201/b16113, 2013

    work page doi:10.1201/b16113 2013

  29. [29]

    Z. Liu, H. Qiao, Semi-group S-System, Science Press, Beijing, 2008, (in Chinese)

    work page 2008

  30. [30]

    J. Lu, H. Li, Y . Liu, F. Li, Survey on semi-tensor product method with its applications in logical networks and other finite-value d systems, IET Contr . Theory Appl., V ol. 11, No. 13, 2040-2047, 2017

    work page 2040

  31. [31]

    Lund, The tensor t-product: A definition for function s of third-order tensors, Numeric

    K. Lund, The tensor t-product: A definition for function s of third-order tensors, Numeric. Linear Alg. Appl. , V ol. 27, No. 3, e2288, 2020

    work page 2020

  32. [32]

    Y . Na, Z. Liu, R. Gao, Predicting multiple types of micro rna-diserse associationos based on tensor factorization and label prop agation, Comp. Biol. Medic. , V ol. 146, 105558, 2022

    work page 2022

  33. [33]

    Tsai, Contributions to the Design and Analysis of Nonlinear Model s, Ph

    C. Tsai, Contributions to the Design and Analysis of Nonlinear Model s, Ph. D. thesis, Univ. of Minisota, 1983

    work page 1983

  34. [34]

    Applied Math

    B.Wei, Second moments of LS estimate of nonlinear regre ssive model, J. Applied Math. in Chinese Universities , V ol. 1, No. 2, 279-285, (in Chinese), 1986

    work page 1986

  35. [35]

    Wang, Parameter Estimate of Nonlinear Models: Theory and Appli- cations, Wuhan Univ

    X. Wang, Parameter Estimate of Nonlinear Models: Theory and Appli- cations, Wuhan Univ. Press, Whhan, 2002

    work page 2002

  36. [36]

    Z. Wang, F. Zhang, Z. Wang, Reserach on return supply cha in super- network model based on variational inequalities, Proc. 2007 IEEE Int. Conf. Aut. Logist. , 25-30, 2007

    work page 2007

  37. [37]

    D. Xie, H. Peng, L. Li, Y . Y ang, Semi-tensor compressed s ensing, Dig. Sign. Proc. , V ol. 58, 85-92, 2016

    work page 2016

  38. [38]

    Y ang, J

    Y . Y ang, J. Zhang, A study on nonnegative tubal matrices , arXiv:2107.11727v2, 2021

    work page arXiv 2021

  39. [39]

    Y . Y an, D. Cheng, J. Feng, H. Li, J. Y ue, Survey on applica tions of algebraic state space theory of logical systems to finite sta te machines, Sci. China Inf. Sci. , https://doi.org/10.1007/s11432-22-3538-4, 2022

    work page doi:10.1007/s11432-22-3538-4 2022

  40. [40]

    Zhang, S

    Z. Zhang, S. Aeron, Exact tensor completion using t-svd , IEEE Trans. Signal Proc. , V ol. 65, No. 6, 1511-1526, 2016

    work page 2016

  41. [41]

    Zhong, D

    J. Zhong, D. Lin, On minimum period of nonlinear feedbac k shift registers in Grain-like structure, IEEE Trans. Inform. Theory , V ol. 64, No. 9, 6429-6442, 2018. APPENDIX A. Algebraic Structure related to Cubic Matrices (i) Group: Definition 10.1: Consider (G, ∗), where G is a set with a mapping ∗ : G × G → G. – (G, ∗) is called a semi-group, if a ∗ (b ∗ ...

    work page 2018