After a brief review on the history of semi-tensor product (STP) of matrices, this survey paper introduces general definitions of STP and semi-tensor addition (STA), and the exploring researches on the mathematical essence of STP and STA, including three major branches: Modern Algebra, Geometry, and Analysis. The STP and STA, as cross-dimensional operators, enhance certain developments in classical mathematics, which is basically of fixed dimensions. As a survey paper, it mainly introduces fundamental concepts and basic results with few predictions. We hope to show such a fact that since the STP breaks the dimension barrier of matrix product, it will inevitably cause impact on the classical mathematics, which is of fixed dimensions.
[1] CHENG D.Semi-tensor product of matrices and its application to Morgan′s problem[J]. Science in China,Series F: Information Sciences,2001,44(3): 195212.
[2] CHENG D,QI H,Semi-tensor Product of Matrices-Theory and Application[M]. 2nd Ed,Beijing: Science Press, 2011.
[3] 梅生伟,刘峰,薛安成. 电力系统暂态分析中似的确半张量积方法[M]. 北京:清华大学出版社,2010.
[4] FORNASINI E,VALCHER M E. Recent developments in Boolean networks control[J]. J Contr Dec,2016, 13(1): 118.
[5] MUHAMMAD A,RUSHDI A,GHALEB F A M. A tutorial exposition of semi-tensor products of matrices with a stress on their representation of Boolean function[J]. JKAU Comp Sci,2016,5:330.
[6] AKUTSU T. Algorithms for Analysis,Inference,and Control of Boolean Networks[M]. New Jersey:World Scientific,2018.
[7] VLAD S E. Boolean Systems-Topics in Asynchronicity[M]. London:Elsevier,2023.
[8] CHENG D,QI H,LI Z. Analysis and Control of Boolean Networks-a Semi-tensor Product Approach[M]. London: Springer,2011.
[9] CHENG D,QI H,ZHAO Y. An Introduction to Semi-tensor Product of Matrices and Its Applications[M]. Singapore: World Scientific,2012.
[10] 程代展,齐洪胜,贺风华. 有限集上的映射与动态过程矩阵半张量积方法[M]. 北京:科学出版社,2016.
[11] CHENG D. From Dimension-Free Matrir Theory to Cross-Dimensional Dynamic Systems[M]. United Kindom:Elsevier,2019.
[12] 程代展,齐洪胜. 基本理论与多线性运算[M]. 矩阵半张量积讲义:卷1, 北京:科学出版社, 2020.
[13] 程代展,齐洪胜. 逻辑动态系统的分析与控制[M]. 矩阵半张量积讲义:卷2, 北京:科学出版社,2020.
[14] 程代展,李长喜,郝亚琦,等. 有限博弈的矩阵半张量积方法[M]. 矩阵半张量积讲义:卷3, 北京:科学出版社,2022.
[15] 程代展,纪政平. 有限与泛维动态系统[M]. 矩阵半张量积讲义:卷4, 北京:科学出版社,2023.
[16] LI H,ZHAO G,GUO P. Analysis and Control of Finite-Valued Systems[M]. Boca Raton, USA: CRC Press,2018.
[17] LIU Y,LIU J,SUN L. Sampled-date Control of Logical Networks[M]. Singapore:High Education Press,2023.
[18] ZHANG K,ZHANG L,XIE L. Discrete-Space Dynamic Systems[M]. New York:Springer,2020.
[19] 冯俊娥. 矩阵半张量积入门[M]. 济南:山东大学出版社,2024.
[20] 程代展,冯俊娥,钟江华,等. 工程及其他系统的应用[M]. 矩阵半张量积讲义:卷5, 北京:科学出版社,2024.
[21] LU J,LI H,LIU Y,et al. Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems[J]. IET Contr Thmé Appl,2017, 11(13): 20402047.
[22] CHENG D,WU Y,ZHAO G,et al. A comprehensive survey on STP approach to finite games[J]. J Sys Sci Compl,2021, 34(5): 16661680.
[23] LI H,ZHAO G,MENG M,et al. A survey on applications of semi-tensor product method in engineering[J]. Science China,2018,61(1): 2844.
[24] YAN Y,CHENG D,FENG J, et al. Survey on applications of algebraic state space theory of logical systems to fnite state machines[J]. Sci China,2023, 66(1): 111201.
[25] ZHONG J,LIN D. On minimum period of nonlinear feedback shift registers in Grain-like structure[J]. IEEE Trans Inform Theory,2018,64(9): 64296442.
[26] XIE D,PENG H,LI L,et al. Semi-tensor compressed sensing[J]. Digital Signal Proc,2016,58:8592.
[27] ABRAHAM R,MARDEN J E.Foundations of Mechanics[M]. 2nd Ed. London: Benjamin/Cummings Pub,1978.
[28] KELLEY J L.General Topology[M]. New York:Springer-verlag,1955.
[29] SCHAEFER H H. Topological Vector Spaces[M]. New York:Springer-verlag,1970.
[30] TAYLOR A E,LAY D C. Introduction to Functional Analysis[M]. 2nd Ed. New York:John Wiley & Sons,1980.
[31] ZHANG K,JOHANSSON K. Long-term behavior of cross-dimensional linear dynamic systems[C]//The 37th Chinese Control Conference, Wuhan, China: IEEE, 2018.
[32] CHEN X,CHENG D,FENG J. Matrix projection and its application in image processing[J]. Digital Signal Proc, 2025, 163:105231.
[33] CHEN X,LI Y,ZHANG L,et al. Support vector machinei1handling missing data: a Cheng projection method[DB/OL].[20250401]. https://sysmath.cjoe.ac.cn/issc/EN/abstract/abstract54767.
[34] CHENG D. Cross-dimensional mathematics-a mathematical foundation for STP/STA[DB/OL].[20250401]. http://arxiv.org/abs/2406.12920.
[35] HUNGERFORD T W. Algebra[M]. New York:Springer-Verlag,1974.
[36] HAL B C. Lie Groups, Lie Algebras,and Representations-an Elementary Introduction[M]. New York: Springer-Verlag,2003.
[37] CHEN C. Thenor-Based Dynamical Systems-Theory and Applications[M]. Switzerland:Springer,2024.
[38] LIM L.Tensors and Hypermatrices[EB/OL].[20250401]. http://www.stat.uchicago.edu/~lekheng/work/hla.pdf.
[39] VASWANI A,SHAZEER M,PARMAR N,et al. Attention is all you need[DB/OL].[20250401]. http://arxiv.org/abs/1706.03762.
