A companion to S.Lang's Algebra 4ed. by Bergman G.M.

By Bergman G.M.

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Example text

One can compose various of these maps, but not every map is composable with every other, so one does not have a composition operation S × S → S, but rather, a family of 8 operations, Sjk × Sij → Sik (i, j, k ∈{0,1}). One easily sees that these 8 operations satisfy 16 associativity laws, corresponding to multiplication on the 16 sets Sjk × Sij × Shi , and that there are 2 identity elements, id0 ∈S00 and id1 ∈S11 . One might abstract this situation, and call a family of four sets S00 , S01 , S10 , S11 given with operations satisfying the appropriate conditions a ‘‘bimonoid’’.

Hence Lang, and many other authors, parsimoniously define the objects of their auxiliary category simply to be these maps g. Lang then forgets to mention that he is redefining the ‘‘free group on S ’’ to mean the object of this category which we would call (Fab (S ), f ), and which he simply writes f . Once this redefinition is noted, what he says is correct. In fact, he will not be consistent – he will sometimes speak of the free abelian group as a map (as above), sometimes as a group given with such a map, and sometimes simply as a group.

But rather than calling this set A ° B and putting a group structure on it, let us call it X, and construct a group of permutations of X. G. M. 43 This group will be generated by the homomorphic images of A and B corresponding to an action of each of these groups on X. The goal is that (a1 , ... , an ) ∈X should be the sequence arising by starting with the empty sequence () ∈X, applying an to this, then an –1 to the result, etc.. This tells us how we ought to define certain cases of the actions of A and B on X, namely (i) If (a1 , ...

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