A Treatise on Universal Algebra Volume 1; With Applications (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1898 Excerpt: ...and it follows that (cpc ) = 0. Hence.c.-.c are mutually normal. (4) Let Ci', Ci," etc., be other points in the latent region of the root 71, so that C' = 7, Ci', etc.: then the same proof shows that c/ is normal to all of c2, ... c, and so on. Hence the latent region corresponding to 7 is normal to the latent region corresponding to p3, and so on. (5) In the same way it can be proved that the whole semi-latent region corresponding to any latent root 7 is normal to the whole semi-latent region corresponding to any other latent root y2. For let di be any point in the semi-latent region of 7 of the first species. Then fai = 7 + X, Ci, fa? = 7.A. Hence (cj fa) = 7 c? dj), by (3) and (4). Also (d, J (fa) = 72 (d, I Cs) = 7, (c.,1 d, ). But (cj ldj) = (dj I c2), by hypothesis. Hence (7, --72) (cj d, ) = 0, and 71+73 ty hypothesis. Therefore (qj d, )=0. Hence the semi-latent region of the first species corresponding to 71 is normal to the latent region corresponding to 72. Similarly the semi-latent region of the first species corresponding to 7.J is normal to the latent region corresponding to 7, . Again dj and dj lying respectively in the semi-latent regions of the first species corresponding respectively to 7, and to 72 are normal to each other. For (dj I fa2) = (d, (7 + X)) = 72 (da I d, ), and (d, fa) = 7, (dj d, ). Thus (dj I fa) = (da I fa) gives (7 -72) (dj d?) = 0; and hence (dj d, ) = 0. Similarly if /i be another point in the semi-latent region of the second species of the root 7, such that //j = ifi+ihdi, then the same proof shows that/i is normal to a, d2 and/2; and so on. Hence the semi-latent regions of different roots are mutually normal. (6) Again consider the equation (c14d1) = (d, fc1). This becomes 7x (d d, ) + Xj (cj d)