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Extra info for Algebra Vol. I

Example text

As regards multiplication of integers, the number 1 is obviously the unity element for which we retain the normal notation. We shall sometimes denote the unity element of a set by 1 even when it is not the number I provided that it will give rise to no misunderstanding. If there is such a possibility then we may use the symbol 1. > of even numbers, let the customary multiplication be defined. ) Since for even numbers a, b (0 0) ab ¢ b, there is no unity element in this set. In the set of mappings of a set into itself, obviously the identical mapping is the unity element.

Since in a o /I the elements a, fi are variables, we can call our compositions "com- positions of two variables". Similarly, one may speak of "compositions of several variables", although we shall not discuss these. We shall, however, deal with "compositions of one variable". We shall usually denote compositions by + or . The composition is then called addition or multiplication, respectively. In these cases we say that the additive, or multiplicative notation is used for the composition. We call a + j9 the sum and a 9 the product of a and f, where a, # are the terms of the sum a + fi or the factors of the product a fl.

A further important example is the product a,B of two mappings a, ,B of a set into itself, as defined in § 4. We shall generalize these and similar well-known examples below. >, if an element of S is uniquely assigned to each ordered pair a, j9 of elements of S in a certain way. We denote this element by a 0 j3 where 0 is the sign of composition. This may then be called the "composition 0". Alternatively, the composition is a mapping of the product set S X S into S or a functionfloc, fi) defined in S.