Theory of determinants. Up to 1841 by Muir T. PDF

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Example text

En e1 e2 s3 . . sn and similarly s1 s2 . . sn = (s1 e2 . . en )(e1 s2 e3 . . en )(e1 e2 s3 e4 . . en ) . . (e1 . . en−1 sn ). Now we set si = e1 e2 . . ei−1 si ei+1 . . en . Then s1 s2 . . sn = s1 s2 . . sn and si ∈ Hi (e) for every i = 1, 2, . . , n. 2 Internal Spined Products Let S be a semigroup and φ a homomorphism of S onto Q. Suppose H1 and H2 are subsemigroups of S such that φ(H1 ) = φ(H2 ) = Q. If the external spined product H1 Q H2 over Q with respect to φ| H1 and φ| H2 is isomorphic to S under the mapping (h 1 , h 2 ) → h 1 h 2 where (h 1 , h 2 ) ∈ H1 Q H2 , then S is said to be the internal spined product of H1 and H2 over Q.

Suppose M and N are normal sub-orthocryptogroups of S. Take m ∈ M and n ∈ N . Note that S satisfies the Eqs. 3). Then we have mn = mn(mn)0 = mnm −1 mn 0 ∈ N M since mnm −1 ∈ N and n 0 ∈ M. Thus, M N ⊂ N M and vice versa. Hence, M N = N M. Then (M N )(M N ) = M M N N = M N and so M N is closed under multiplication. Next we take m ∈ M and n ∈ N . We have (mn)−1 = m 0 n −1 m −1 n 0 ∈ M N M N = M N . Hence, M N is closed under taking inverse. Since M and N are full, M ⊂ M E(S) ⊂ M N and N ⊂ E(S)N ⊂ M N .

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Theory of determinants. Up to 1841 by Muir T.


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