# 2-uniform congruences in majority algebras and a closure by Czedli G.

By Czedli G.

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Z2) = :22[s4,s7,s12J/<2S7>. 2s 7 = 0 for integral coefficients, the same holds fOr:2 2 Since coefficients. In the mod 2 Bockstein spectral sequence we have B (Y4) = 0 for all k and k Y7' o 2 Since P2(s12) = Y6 and s12 is torsion free, it follows that for all k. o for all k. Then the 2-torsion elements in 2H *(BG ) 2 2 torsion free elements correspond to

2 is determined. well. 1 x 55 ; LZ/~) From the fact that by LZ/£[u ,v J, then u 2 2 0* .... [qJu 2 and By the commutativity of 0, in (4:2), (¢qx 2 )* Since i* is a monomorphism, ¢q* Explicitly ¢. (Y4) 2 2 [qJ Y4 and l' q* B (Y12) B is determined as + [q J 6 Y12· We claim that the hypotheses of (Smith's) Theorem 3-1 hold for the Cartesian square P B (4:3) B ----. B x B which we have extracted from diagram (4:1) above. Since G2(~) is connected, the construction insures that B x B is simply connected.

Then everything in section 4 goes through except for the fact that E 2 = Eoo of the Eilenberg-Moore spectral sequence may not be the true algebra structure for £ = 2. ) 2 (For £ = 2, x 2i + l o is a genuine This proves the first part of Theorem 1-9. We now want to calculate 2 H*(BG (W )). 2 q As with the primes different from two, we will want to consider diagram (5:1) with ~2-coefficients. It is first necessary to examine P2' the reduction mod-2 map. Recall, from section 5, that there are elements t 3 , tIl E H*(G 2 ;2Z) Y4' since x , x x ' and Y4 are the unique non-zero elements in 3 5 6 their respective dimensions.

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