By Redei L.
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Additional info for Algebra Vol. I
If an addition and a multiplication are defined in a set S, and S has a zero element 0, then Oa=a0=0 is always valid. Since 0 + 0 = 0, (0 + 0)a = 0a , a(0 + 0) = a0 . 15) Hence it follows from the distributivity that Oa+Oa=Oa, a0+a0=a0. 15) follows from the regularity of the addition. We repeatedly emphasize that the type of notation used for compositions is of no importance. Accordingly we may write non-commutative and non-regular compositions in the additive notation, but not without pointing it out.
Alternatively, the composition is a mapping of the product set S X S into S or a functionfloc, fi) defined in S. However, since the above concept of compositions is too general, we shall only consider in detail compositions with certain special properties. 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".
If, on the contrary, x < y denotes that x is a child of y, there is no semiordered set owing to the absence of transitivity. ) of the natural numbers and also in the set of the real numbers, then the usual relation < denotes an ordering relation. In these sets the sign < retains its former meaning. If, on the contrary, we wish to define another ordering or semiordering relation in one of these sets, then we denote it by the sign -<. This applies to other analogous cases. Similarly the terms "less" and "greater" are not used when they might give rise to a misunderstanding.
Algebra Vol. I by Redei L.