Cross-Dimensional Mathematics: A Foundation For STP/STA

TL;DR

Cross-Dimensional Mathematics (CDM) provides a unified foundation for semi-tensor product and semi-tensor addition by extending classical algebra and geometry to objects with mixed dimensions. It builds three layers—hyper algebra, hyper geometry, and hyper Lie theory—based on MVMDs, introducing hyper groups, hyper rings, hyper modules, hyper vector spaces, and hyper manifolds, including a Lie theory for non-square matrices. The framework defines dimension-keeping and pseudo-stretch operations, cross-dimensional permutation groups, and a rich equivalence/identity structure that yields quotient constructions and component-wise correspondences with traditional algebra. It also develops metric, differentiable, and manifold structures on ${ m R}^{ty}$, together with a cross-dimensional generalization of general linear groups and algebras, thereby enabling cross-dimensional control, geometry, and dynamics with potential applications to dimension-varying systems and multi-scale modeling.

Abstract

A new mathematical structure, called the cross-dimensional mathematics (CDM), is proposed. The CDM considered in this paper consists of three parts: hyper algebra, hyper geometry, and hyper Lie group/Lie algebra. Hyper algebra proposes some new algebraic structures such as hyper group, hyper ring, and hyper module over matrices and vectors with mixed dimensions (MVMDs). They have sets of classical groups, rings, and modules as their components and cross-dimensional connections among their components. Their basic properties are investigated. Hyper geometry starts from mixed dimensional Euclidian space, and hyper vector space. Then the hyper topological vector space, hyper inner product space, and hyper manifold are constructed. They have a joined cross-dimensional geometric structure. Finally, hyper metric space, topological hyper group and hyper Lie algebra are built gradually, and finally, the corresponding hyper Lie group is introduced. All these concepts are built over MVMDs, and to reach our purpose in addition to existing semi-tensor products (STPs) and semi-tensor additions (STAs), a couple of most general STP and STA are introduced. Some existing structures/results about STPs/STAs have also been resumed and integrated into this CDM.

Figures (10)

• Figure 1: Cross-Dimensional Mathematics vs. Existing Math.
• Figure 2: Hasse Diagram
• Figure 3: Group Decomposition of a Hyper Group ${\mathcal{G}}$
• Figure 4: Sub Hyper Groups
• Figure 5: Ring Decomposition of a hyper ring ${\mathcal{R}}$

Theorems & Definitions (202)

• Definition 1.1
• Definition 2.1
• Example 2.2
• Definition 2.3
• Definition 2.4
• Example 2.5
• Definition 2.6
• Example 2.7
• Definition 2.8
• Example 2.9