By Mazurov V.D.

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42 11. DIRECT SUMS If K is an ideal of the Boolean algebra B of all subsets of 1, then by the K-direct sum of the Bi we mean the subset of Bi whose elements are the vectors a with s(a) E K. Since s(al - a z ) s s(aJ u s(a,), the K-direct sum is a subgroup of B i ; it will be denoted by the symbol n n OKBi. If, in particular, K consists of all finite subsets of 1, then we get the direct sum, while if K = B, then we arrive at the direct product. Among the subgroups of the direct product there is an important type which frequently occurs in algebra.

Y, . . satisfying the following axioms: I . With each ordered pair A, B of objects in %? there is associated a set Map(A, B) of morphisms in %? such that every morphism in %? belongs to exactly one Map ( A , B). If a E %‘ belongs to Map(A, B) then we write a : A + B and may call a a map of A into B, while A is the domain, B the range of a . 2. With a E Map(A, B) and j? E Map(B, C ) , there is associated a unique element of Map(A, C ) , called their product Pa. 3. Whenever the products are defined, associativity prevails: 4.

I i For a group G, we introduce two maps: the diagonal map AG : G [the number of components can be arbitrary] as A,: g - ( . . ) -+ n G ( 9 E GI, , and the codiagonal map V , : 0 G + G as V G : ( . * , g i , * * * ) ~ C (ggii E G ) . i If there is no danger of confusion, we may suppress the index G. 3. 211+ B , Wz@Pz)A ’B 3 @ c2 and V(UZf€JPl) A, OBI ’B2 v z P 2 * C 3 + 0 . 3 Since A, and a, are monic, so is a, Al. , a, = O and p, a, 1, = p , p1a1= 0. If b, E B, belongs to Ker(p, @ P2)A, then both p 2 b 2= 0 and P2b2= O .