  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.
