Rows have a set of operations which we can use on them; in mathematical terms, they form a group. The most important operation is that of row multiplication or transposition - in this operation, the bells in one row are rearranged according to the order given by another row.
For example,
21345678 * 13572468 = 23571468
and
13572468 * 21345678 = 31572468
Note that the two above examples do not give the same result; that is, the order in which two things are multiplied does matter.
As an example of how row multiplication is used, suppose that we want to know the 4th row of the lead of Plain Bob Major with lead head 17856342. The 4th row of the first lead (which has a lead head of rounds) is 42618375. Multiplying these together gives us the answer we are looking for, namely 57312846.
The identity for this operation is rounds; in other words, any row multiplied by rounds gives itself, and rounds multiplied by any row gives that row.
It is possible to define the inverse of a row as the row which, when multiplied by that row, will give rounds. For example, the inverse of 13572468 is 15263748, as
13572468 * 15263748 = 12345678
The opposite of row multiplication is row division. If a * b = c, we can define c / b = a. Using the same example as above, suppose we have a lead of Plain Bob Major and we know that the fourth row is 57312846, and we wish to find the lead head. Just divide by the fourth row of the plain course (42618375) to get the answer.