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

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The latter assertion follows from e' = e'e" — e'\ which proves Theorem 24. An element 0(6 F) is called a zero element of F, if 0a = aO = 0 (for all a 6 F) . 8) THEOREM 25. A semigroup contains at most one zero element. If both 0 and 0' are zero elements in F, it follows from the definition that 0 = 00' = 0', which proves Theorem 25. For the sake of clarity the following should be noted. 8) in semigroups, the latter might have been denoted "multiplicative zero element". This is unnecessary as no confusion can arise except in the case of a structure S with two operations; but here (by virtue of distributivity) Theorem 22 holds, so that it follows that the zero element of S x agrees with that of S+.

If, in a set © of subsets of a given set, there is to each chain of sets an upper bound {in ©), then © has a maximal element. ) We may briefly say that Theorem 15 refers to semiordered sets, and Theorem 16 to sets of sets. When applying them, we shall refer to each as "the KURATOWSKI-ZORN lemma" since Theorems 15 and 16 are in fact two equivalent propositions as is shown in the following paragraphs and may also be proved directly. § 13. The Lemma of Teichmiiller—Tukey THEOREM 17 (lemma of TEICHMULLER—TUKEY).

Except for the zero element there are no elements simultaneously nilpotent and idempotent. For an a (6 F) and a natural number n we denote by %/OL the solution of the equation |" = a and call it an nth radical of a. According to this, ^fi is in general a manyvalued symbol. If the equation has no solution, we say that ^/a is non-exist ent. When n = 1, then^/a = a, therefore we usually assume n ^ 2. yet is also called the nth root of a and in the cases n = 2, 3 a square root, or cube root, of a respectively.

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