TITLE: Is it possible to exist a "three-dimensional matrix"?
QUESTION [6 upvotes]: We all have seen matrices that are "bi-dimensional" (i'm using the quotes here because i'm not talking about the number of lines, but about the way you represent a matrix, as a rectangle).
I was wondering if we can define a space where we have similar objects, but instead of being a rectangle it would be a box, cubic or non-cubic.  Imagine for a example a space $M$ of "3D" Matrices, and a 3D matrix $A$ would have numbers represented by $a_{ijk}$, where $i$ is equivalent to a column, $j$ to a line, and $k$ to another line, but this time in a 3-dimensional way. We can imagine a simple $2\times2\times2$ "3D-Matrix" we would have eight entries: $a_{111}, a_{112}, a_{121}, a_{122}, a_{211}, a_{212}, a_{222}$ instead of 4 in a normal matrix.
Does this concept exist? I have tried looking in this forum for "3D matrices" but didn't found anything. Also looked on google but couldn't find anything. Thanks.

REPLY [3 votes]: In combinatorial circles I have heard these referred to as "hypermatrices". You can define binary addition and an $n$-ary multiplication(where $n$ is the number of dimensions of the hypermatrix.) that is a natural analogue of matrix multiplication. 
Unfortunately you lose a lot of theory because, for example, I don't think anyone has found a natural analogue to eigenvalues or many other concepts natural to the study of matrices.