By D. Arnold, R. Hunter, E. Walker

ISBN-10: 3540084479

ISBN-13: 9783540084471

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In this case S T is a monoid with identity (0S , 1T ). Here by commuting actions we mean (ts)t = t(st ) for all s ∈ S and t, t ∈ T . Again, if S is written multiplicatively, then we shall use exponential notation for the actions of T on S. It is often convenient, and conceptually useful, to view the multiplication formula for the two-sided semidirect product as an instance of matrix multiplication. That is, if we identify (s, t) with the matrix t0 , then we have the formula: st t0 st t 0 s t = tt 0 st + ts tt .

In all other areas of mathematics is used for the wreath product, so we use this symbol, as well. The following series of exercises establishes some well-known properties of wreath products [85]. 3. Verify that (X, S) (Y, T ) is a well-defined right transformation semigroup. 4. Show that if (X, S) and (Y, T ) are faithful, then so is (X, S) (Y, T ). 5. Show that if (X, M ) and (Y, N ) are right transformation monoids, then (X, M ) (Y, N ) is a transformation monoid. 6. Show that if (X, G) and (Y, H) are right transformation groups, then (X, G) (Y, H) is a transformation group.

36) One sometimes calls f1 ×T f2 the pullback of f1 and f2 (along T ). For homomorphisms, things reduce to the usual notion of pullbacks. Now we turn to some examples that will be important later on. Let f : S → T and g : U → V be relational morphisms. Consider f : S → T × V and g : U → T × V given by −1 f = f p−1 T and g = gpV . Then one easily checks: f × g = f ×(T ×V ) g . 37) Let f : S → T be a relational morphism. Observe that S ×f,1T T = {(s, t) ∈ S × T | t ∈ sf } = #f and f ×T 1T = pT . 38) More generally, f ×T g is the graph of the relation f g −1 .

### Abelian Group Theory by D. Arnold, R. Hunter, E. Walker

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