Cross-Dimensional Mathematics: A Foundation For STP/STA

TL;DR

This paper introduces cross-dimensional mathematics (CDM), a new framework combining hyper algebra, hyper geometry, and hyper Lie groups to handle mixed-dimensional structures, extending semi-tensor products and additions.

Contribution

The paper develops the foundational theory of CDM, integrating hyper algebraic and geometric structures with generalized semi-tensor operations for mixed-dimensional spaces.

Findings

Defined hyper algebraic structures like hyper groups, rings, and modules.

Constructed hyper geometric spaces including hyper manifolds and hyper metric spaces.

Introduced generalized semi-tensor product and addition extending existing concepts.

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.