 By Rédei, L.; Sneddon, I. N.; Stark, M

ISBN-10: 1483197611

ISBN-13: 9781483197616

ISBN-10: 1483222640

ISBN-13: 9781483222646

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Additional resources for Algebra Volume 1

Sample text

It should be noted that two different mappings are always different as operators. An operator a is called a null operator, or identical operator, if aa = 0 or aa = a, respectively, for all a. Of course, several null operators and identical operators may occur in 0. § 19. Structures A non-empty set S =< is called an algebraic structure, if at least one composition (or more exactly an ordered rc-tuple of compositions) is defined in it. We say that S constitutes a structure with these composi­ tions.

Some­ times we use ^/a to denote the set of all solutions of the above equation, but this must be pointed out whenever used. An equation a = ax . aB (or the right-hand side of it) is called a factor decomposition of a where we mostly suppose n ^> 2. 50 STRUCTURES If for two elements a, fl the equation af = j8 is solvable, then we say that a is a left divisor of /? or that ft is a left multiple of a, and we write oc|/?. We also say that /? // divisible by a. (For r/#fe divisibility, we introduce no notation.

Cf. SZELE (1949—50) as to the above proof and the following sections. § 12. The Special Lemma of Kuratowski—Zorn As mentioned above, c denotes in each set © of subsets of a set a semiordering relation. From now on any set © will be regarded as semiordered (according to this relation c:). In particular if the set © is ordered, then it is called a chain of sets. Accordingly, this means a set of sets, where for any two different elements A, B either A a B or B a A holds. It is usual to call an ordered subset of an arbitrary semiordered set a chain.