By Rédei, L.; Sneddon, I. N.; Stark, M
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Additional info for Algebra Volume 1
Now, it is important in connection with multiplication that associativity, invertibility, regularity, commutativity and distributivity are self-dual properties. 1) where S denotes the structure under consideration. By dualization we obtain from it: y05«) = (ypfr . 2) are essentially identical. This shows that associativity is self-dual. 3) are solvable for each pair a, /?. 3) after dualization (and the interchange of £ and rj) themselves interchange, it is clear that invertibility is self-dual.
Aw are distinct and every term in the sequence is equal to one of these n terms. If a, a 2 ,. . are not all different then there is a unique natural number n such that a , . . , a" are all different and The product of two products / ii ax . . a,, px . . f}k is equal to <*i- • . < * / & . . & . We repeat that the unity element s is defined by the property £a = ae = a (a 6 F) . 7) holds. Consequently, by dualization a right unity element e" is characterized by the property oce" = a (a € F). THEOREM 24. Iff has no unity element, ^6« e/fAer f/ze se£ o/fe/if unities or the set of right unities is empty. It follows by dualization from the first part of the theorem that if there exists a unity element then F has no further right unity element.
Algebra Volume 1 by Rédei, L.; Sneddon, I. N.; Stark, M
The product of two products / ii ax . . a,, px . . f}k is equal to <*i- • . < * / & . . & . We repeat that the unity element s is defined by the property £a = ae = a (a 6 F) . 7) holds. Consequently, by dualization a right unity element e" is characterized by the property oce" = a (a € F). THEOREM 24. Iff has no unity element, ^6« e/fAer f/ze se£ o/fe/if unities or the set of right unities is empty. It follows by dualization from the first part of the theorem that if there exists a unity element then F has no further right unity element.