# A basic relation between invariants of matrices under the by Rogora E.

By Rogora E.

The 1st primary theorem of invariant idea for the motion of the designated orthogonal workforce onm tuples of matrices via simultaneous conjugation is proved in [2]. during this paper, as a primary step towards developing the second one basic theorem, we research a easy identification among SO(n, okay) invariants ofm matrices.

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**Sample text**

If a E supp(f + g) then o:/; I(a) + g(a) and so either I(a) :/; 0 or g(a) :/; 0, which is to say that a E supp(f) U supp(g). Conversely, suppose that a E supp(f) U supp(g). Then either I(a) :/; 0 or g(a) :/; 0 and so, since R is zerosumfree, (f + g)(a) = I(a) + g(a) :/; o. 0 If 0 :/; B ~ A then any 1 ERA can be written as 1 = It +12, where supp(ft} ~ Band supp(12) ~ A \B. Note too that if R is zerosumfree and if I,g E RA satisfy 1 ~ 9 then supp(f) ~ supp(g). ,g): a o-t I(a) { 0 If R is additively idempotent then I""g if a ~ supp(g) otherwise.

If A is a nonempty set then the semiring RA is a natural setting for the study of generalizations of operator theory, known collectively as "idempotent analysis". See [272] for details. (V) EXAMPLE. It is often the case that we look at products of the form RA, where R is a semiring and A is a partially-ordered set. In this case the elements of RA are R-valued pomsubsets (= partially-ordered multisubsets) of A and if A has a total order defined on it then the elements of RA are R-valued tomsubsets (= totally-ordered multisubsets) of A.

If a, e E A then 2: [2: bEA h(a,X)k(X,b)] 2: rEA = [2: h (b,Y)k(Y,e)] yEA [(h k)(a, b)] [(h 0 0 k)(b, e)] bEA = [(h 0 k) 0 2: = 2: [2: (k 0 h)](a, e) = [(hoh)o(kok)](a,e) = [(h 0 h)(a, b)] [(k 0 k)(b, e)] bEA bEA as desired. h(a, x)h(x, b)] xEA [2: k(b, y)k(y, e)] yEA 0 If we consider the particular case in which R is a CLO-semiring and A, B, and Care nonempty sets then there are other possible, and useful, ways of defining compositions between relations h E R AxB and k E R BxC . Some of these, along with their applications, were considered in [28, 30] and then extended in [75] for the case of R being (IT, V, 1\).