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The natural embedding mod 2 + mod A corresponds to the extension n H no of Smodules to 9’ by 0, and Ext$(s-, no) is identified with Hom(s” 15, n) =: N(n). Finally, F corresponds to an isomorphism of categories mod Y G Nk, m - (m(s),f, ml-V, where f is induced by the structure maps m(s) 0 9’(t, s) -+ m(t), t E Y. For instance, let 9’ = { pl, . . 9. The Smodules py then form a spectroid of mod 5 Since dim N(pY) = dim DY(pi, s) = 1, we infer that mod Y ; Nk is equivalent to Mqk and (rep 9)““. Remark.

Finitely Represented Posets g’ = g 3 b,. If b, E @” and b, 4 g’, case 2 provides the contradiction ~c~+u{a,}~b,. If b, +! p 2 (the eventuality g = {u2 > a3 < c1 > c2, b, v c3 > b,} is excluded because 9 cannot contain the critical subset (u2 > ,a3: b, > b,: c2 > c,}). Finally, if {b, , b2} c g+, 8’ + 3 u 2 u 1 is not contained in a full subposet of g of the form 3 u2 \ 2 (otherwise (u2 > a3, b, > b,, c2 > c3} would be critical in g). Thus we may apply case 3 to ?? = {u2 < a, > b, < b,, c1 > c2 > cg > cq < u,> ; 4 7 2 T 2.

In this spirit, we want to examine some elementary properties of modules which derive directly from the constraints imposed upon aggregates and spectroids. Let Y be a spectroid: If M E modPf Y is pointwise finite, each endomorphism cp of M gives rise to decompositions M(X) = M(X), @ M(X),, X E Y, where q(X) is nilpotent on M(X), and bijective on M(X), . These punctual decompositions provide a global decomposition M = M, 0 M, such that cpis “pointwise nilpotent” on MO and invertible on M,. In particular, if M is indecomposable, 40must be pointwise nilpotent or invertible.

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A basis of identities of the Lie algebra s(2) over a finite field by Semenov K.N.

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