By Rédei, L.; Sneddon, I. N.; Stark, M
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Additional resources for Algebra Volume 1
According to G.
However, as this is impossible by reason of the definition of R*, s* must be a maximal element of S, proving Theorem 15. In the literature this theorem is usually referred to as ZORN'S lemma, although KURATOWSKI (1922) was the first to enunciate it. ZORN (1935) was the first to apply it to algebra. 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.
A regular element of a semigroup either has an inverse or has no left inverses. The product of two regular elements is regular. All the regular elements of a semigroup F, providing such exist, form a semigroup F*. In a regular semigroup it follows from each equation SOL = a that e is the unity element, The semigroup F* of this theorem will always be denoted by an asterisk and called the regular semigroup of F. ) In order to prove the theorem we first note that its first part may be stated as follows: in a semigroup F a left inverse of a regular element must be at the same time the inverse of this element.
Algebra Volume 1 by Rédei, L.; Sneddon, I. N.; Stark, M