When we look at the operations in the Grzegorczyk hierarchy, we see that all ones after multiplication are not commutative. Thus it is worth thinking about what should we do to derive an expression like c"c"..."c (non associative) from a set of equal numbers c connected with an operation '. Operation ' is the previous for ", i.e., c'c'c'c'c = c"5.

How to achieve the result like c"c"c"...?

First of all, we must organize this set A = {c, ..., c} of equal numbers c.

We can order them in three ways.

1) To make a multidimensional set, a matrix in which each element has N different independent indexes i1, i2,..., iN.

2) When this hypermatrix is ready, then we can in mind make a similar matrix consisting of the matrix dimensions itselves. I. e., Let's make a matrix which has 15 dimensions. Then we can order them in a flat matrix 3 x 5 (two dimensions). It is the first way to order a hypermatrix.

3) Finally, independently from the previous method, we can take a set of all values of one indexes; for example, a flat matrix has two dimensions, and along one of them an index goes through values 1, 2, 3, ..., N. Then let's make a hypermatrix which is filled with these numbers 1, 2, 3, ..., N. It is the second way to order hypermatrix.

Of curse, we can continue applying these methods; for example, if we have already made a hypermatrix M of dimensions itselves (the 1st way, point (2)), then we can try to make another hypermatrix from the dimensions of this hypermatrix M, and so on, Or we can try to apply immediately all the two ways of extra-ordering.

Of course, we can imagine some other secondary cases of ordering. For example, each element of hypermatrix can itself be a hypernatrix, but in this case one hypermatrix is 'put' into the other, and it is not quite interesting.

So, these ways of ordering we can interprete so.

Let a matrix M have dimensions N1, N2, N3 (each one is a number of elements (rows, columns) along the "direction" relating to it). When we connect all the elements of the matrix with an operation #, and the next oparation is %, then we can shortly write the result as a%(N1 % N2 % N3). Of course, it is not derived, but is put as a useful and obvious definition.

The, when we make additionally a hypermatrix from numbers N1, N2, ... and it has dimensions M1, M2, M3, M4, then instead of numbers N1, N2, N3 in the brackets we can (by definition) write a matrix, in which all elements don't hover in the void but are connected with the operation %.

And so on.

The expression in the brackets includes 3 elements. If we do not order a set of elements of the matrix M (in other word, if we do not make that matrix) then we have a result as a%(N1*N2*N3) where * is usual multiplication. Then in the brackets we see only one natural number N = N1*N2*N3.

So, let's order dimensions

How to achieve the result like c"c"c"...?

First of all, we must organize this set A = {c, ..., c} of equal numbers c.

We can order them in three ways.

1) To make a multidimensional set, a matrix in which each element has N different independent indexes i1, i2,..., iN.

2) When this hypermatrix is ready, then we can in mind make a similar matrix consisting of the matrix dimensions itselves. I. e., Let's make a matrix which has 15 dimensions. Then we can order them in a flat matrix 3 x 5 (two dimensions). It is the first way to order a hypermatrix.

3) Finally, independently from the previous method, we can take a set of all values of one indexes; for example, a flat matrix has two dimensions, and along one of them an index goes through values 1, 2, 3, ..., N. Then let's make a hypermatrix which is filled with these numbers 1, 2, 3, ..., N. It is the second way to order hypermatrix.

Of curse, we can continue applying these methods; for example, if we have already made a hypermatrix M of dimensions itselves (the 1st way, point (2)), then we can try to make another hypermatrix from the dimensions of this hypermatrix M, and so on, Or we can try to apply immediately all the two ways of extra-ordering.

Of course, we can imagine some other secondary cases of ordering. For example, each element of hypermatrix can itself be a hypernatrix, but in this case one hypermatrix is 'put' into the other, and it is not quite interesting.

So, these ways of ordering we can interprete so.

Let a matrix M have dimensions N1, N2, N3 (each one is a number of elements (rows, columns) along the "direction" relating to it). When we connect all the elements of the matrix with an operation #, and the next oparation is %, then we can shortly write the result as a%(N1 % N2 % N3). Of course, it is not derived, but is put as a useful and obvious definition.

The, when we make additionally a hypermatrix from numbers N1, N2, ... and it has dimensions M1, M2, M3, M4, then instead of numbers N1, N2, N3 in the brackets we can (by definition) write a matrix, in which all elements don't hover in the void but are connected with the operation %.

And so on.

The expression in the brackets includes 3 elements. If we do not order a set of elements of the matrix M (in other word, if we do not make that matrix) then we have a result as a%(N1*N2*N3) where * is usual multiplication. Then in the brackets we see only one natural number N = N1*N2*N3.

So, let's order dimensions