## A Treatise on Universal Algebra Volume 1; With Applications

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) ...

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## 1.A Treatise on Universal Algebra Volume 1 With Applications

ISBN 10: 1236078357 ISBN 13: 9781236078353
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