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

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The many-valuedness of ^/a may often be reduced by assigning to ^/a one of its possible values and considering this value consistently. In cases where^/a has but one value or where we have chosen once and for all a value for ^/a , we usually denote this value by a1/rt . 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.

We remark that the axiom of choice is also a conclusion from Theorem 19. If S is a set, we use this theorem to make it well-ordered, then let the minimal element of each (non-empty) subset correspond to that (non-empty) subset of 5, thus creat­ ing a choice function. , the axiom of choice, the lemma of KURATOWSKI—ZORN, the special lemma of KURATOWSKI—ZORN, the lemma of TEICHMULLER—TUKEY, the theorem of HAUSDORFF—BIRKHOFF and the theorem of well-ordering of ZERMELO, are connected in such a way that each of them follows from the previous one, and the first follows from the last.

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.

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Algebra Volume 1 by Rédei, L.; Sneddon, I. N.; Stark, M
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