By Jenca G.

We turn out that if E1 and E2 are a-complete impact algebras such that E1 is an element of E2 and E2 is an element of E1, then E1 and E2 are isomorphic.

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S, , S, . S, ? 5. A = AB I B I. ), 1 -J I AB I BA. ' -j I BA I IAI 6. 7. A = ( O0 1 8. A = (; by 1) 1) 1 0 A=f a 0 0 40 I . VECTORS AND MATRICES 9. of 1 A=(: 10. 11. 1 -1 2 -3 -1 0 -1 : ; 1;) Eq. (64) Ya L ==C by 12. a A=(-b b a) A (AB). a (a of + jb). B b by semigroup have the group property. ) 13. A=( AB # BA -c a f j b jd + a - jb b, b 41 EXERCISES 14. An A by Bn, B n n (AB)" 15. A d a Pn 16. = ad - bc (n + 17. D n. x, dldx. , xn. D A, B, C, n x n (A X D on + B) Y AX+BY=C BXtAY = D X 2n x 2n 18.

3) (x, y) (19) xi*yi = x*Ty = (t) xt = X*T = (xl* (‘11 ‘12 ‘13 = aZ1 a,, ... u31 At hermitian conjugate ... (20) x,*) hermitian conjugate xt A ... x2* x. ’) (21) A. Eq. (3) (X,Y> = xty (23) 3. GENERAL (PROPER) 49 INNER PRODUCT on xty, 3. G E N E R A L (PROPER) I N N E R P R O D U C T (x, x> (A) (B) (x, x) (x, x) x = 0 x = 0 (x, y) y (x, Y> = (x, ay + pz> = a(x, y) + p(x, z ) x, (Y, x > * 44). I). (3) K I, xtKy K = kij , (ytKx)* 8). (25) 50 11. THE INNER PRODUCT on i,j x y, kij K = = k,*, Kt A square matrix K such that it equals its hermitian conjugate Kt) is called hermitian.

T H E INNER PRODUCT En. a (24) (x, y) (34) = x%y S ST = S , S = sii (sii), = sii . S x (A’) y (34) S, x. (x, x) (x,x) (x, (D’) (x, = = x 0 + bz) x) = a