Volume 1 / Part 4 — *173. THE PRODUCT OF THE RELATIONS OF A FIELD (continued).
Reading edition
*173. THE PRODUCT OF THE RELATIONS OF A FIELD (continued).
Summary of *173.
In this number, we shall consider the relation between the domains of relations related by , i.e. we shall consider . This relation bears to a relation analogous to that which bears to . We shall denote it by "." When , is like , and is often more convenient than . When , arranges the multiplicative class of by first differences, taking first differences to mean that the earliest member of for which has the -member earlier than the -member in the -series.
The properties of all result immediately from those of , and offer no difficulty of any kind. The most important of them are:
*173·14.
I.e. if is not null, and no two members of have the same field, then the field of is the product of the fields of . Observe that if .
*173·16.
*173·2.
*173·22.
*173·23.
*173·3.
*173·31.
*173·01.
*173·1.
*173·11.
*173·12.
*173·121.
*173·13.
*173·14.
Dem.
*173·15.
*173·151.
*173·16.
Dem.
*173·161.
*173·17.
Dem.
*173·2.
*173·21.
*173·22.
Dem.
*173·23.
Dem.
*173·24.
Dem.
*173·25.
Dem.
*173·26.
Dem.
*173·27.
Dem.
The following proposition gives a correlator of and when we are given a double correlator of and .
*173·3.
Dem.
*173·31.
*173·32.
Dem.
*173·33.
The above proposition is used in proving the associative law for "" (*174·401).