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

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

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Additional info for Algebra Volume 1

Example text

7) that e'P = e'(af) = (e'a)f = af = j8 and ^ e " - (ija)e" - rj(* e") = r}<* = fi . With /? = s" and /? = e', respectively, we obtain e V = e" = e'. £" = /? it follows that e" = e' is the unity element for which we now write s. Then we take solutions Q, a of the equations (XQ = S , a

A well-ordered non-empty subset R of S is called regular (relative t o / ) , if the minimal element of R i s / ( S ) and every other element a of R is the chosen element of the set of those elements of S, which are greater than all elements of R(a). , the set which con­ sists of the single element f(S). Since, according to this, every element a of a regular subset R of S is uniquely defined by the cut R(a), the following are valid: (a) If Rl9 R2 are regular subsets of S, then i^iC^) = i ^ f e ) is o n t y possible for ax = #2.

41 SEMIGROUPS What has been said in §17 with respect to an associative multiplication, relates to semigroups. We shall now consider a further series of important concepts in semi­ groups. All the following definitions will be independent of associativity; on the other hand, our theorems would lose their validity without the assumption of associativity. 2) QO We illustrate it with examples. The multiplication table « P 7 p p p a y a a V a P y p means aa = ft, a/? = a, etc. But this multiplication is not associative, since (aa)a = = POL = p and a(aa) = a/?

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