Universal Eigenvalue Equations

ing d = f · b. Conversely, if this homomorphic condition holds everywhere, then apply b to both its sides, observe that Cχb = iA and by abstracting c get h = Cχa with a = h · b. Q.D.E. 5.8 Definitions. Keep the notation of 5.4 and let B be the set of bases of α with index X. Consider a family k: B → F(FXA)C, for some C, and a b ∈ B. If for all b′ ∈ B and M :X → A (43) kbM = kb′(M ◦ b′) , then we say that the function kb: FXA → C, as well as family k, are (absolutely) invariant. (In fact, if kb is invariant, then any kb′ is, as it follows from 5.1 through easy passages.) See [38] for a (lone) definition of universal invariance in F. Klein’s synthetic form. We call an invariant function η: FXA → D general if for any invariant family k as above there is a function f : B → FDC, such that kb′ = fb′ · η for all b′ ∈ B. 5.9 Corollary. If χ is an analytic representative, then η = Cχ is a general invariant function. Proof. By 5.6 η = rb−1: FXA → H αα. Hence, η is an invariant function if (43) holds for the k such that ka = ra−1 for all a ∈ B, i.e. if rb′(ηM) = M ◦ b′ for all M :X → A and b′ ∈ B. This is a trivial identity because of our notation of r, η and ◦ in 1.2, 5.9 and 5.4. Also, η is general, since it is one to one. (We can take fa = ka · η−1 for all a ∈ B.) Q.D.E. 5.10 Example. By the preceding corollary any function or predicate of the endomorphism associated to a family M is invariant. Hence, in based algebras all present theory about universal eigenvalue equations concerns invariants only. E.g. by 5.7(A) independence is an invariant predicate. This differs from Universal Algebra, where noninvariant notions are accepted, as we are going to show. In fact, we disprove the invariance of the “C/Ci–independences” as in [15]. These are weaker notions of independence, sometimes [11] and [17] considered akin of independence itself. In a vector space, they express the lack of certain linear dependencies among the elements (columns) of a family M and are equivalent to the independence of M . Here, we recall a couple of them that correspond to conditions (C3) and (10) of [15]. (However, it is easy to see that the next conterexample works even for conditions (C1) and (C2) ibidem.) Given M and α as in 5.8, consider the following two conditions: ⋂ (C↑V ) = C( ⋂ V ) , for all V ⊆ PM (44) and Mx ∈ Cv implies Mx = vx , for all v ⊆ M and x ∈ X , (45)