By P. a. Garcia Sanchez, P. A. Garcia-Sanchez J. C. Rosales

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Xi) = (xl,... ,x,) Q, then the elements mi,. ,m, are linearly independent if and only if the elements f (mi), ... ,f (m,) are linearly independent. The map g : + p , g ( x ) = xQ is an isomorphism of vector spaces and hence it preserves linear independency of sets of vectors. Let M be a subgroup of Zn generated by {mi,. ,m,} with mi = ( m i l , - . ,m,) for al1 i E (1,. , S ) . Let A be the matrix whose rows are mi, ... ,m,. We know that the matrix A is equivalent to a diagonal matrix of the form with d l , ...

Define on S the following binary relation: z3. ( a , b ) 2 ( c , d ) if and only if a+d = b+c. Prove that (a) z is a congmence, (b) (S x S ) / T is a group, 3. FIMTELY GENERATED CANCELLAWE MONOIDS (c) S is isomorphic to a submonoid of (S x S)/T. The group (Sx S)/zis known as the group of quotients of S. As a consequence of this exercise, every cancellative monoid is isomorphic to a submonoid of a group. 9. Prove that a submonoid of Z is reduced if and only if it is not a nontrivial subgroup of Z(equivalently: if a nontrivial submonoid of Z has a unit, then it is a group).

Hence, N3 / N M is isomorphic O to i l 2 X Z2 X z22. Remarks In [9] and [39] the idea of group of quotients (see the exercises of this chapter) is used to prove that every cancellative monoid can be embedded in a group. In this way, every cancellative monoid is a submonoid of a group. Particularly interesting are the submonoids of Zn and W since they represent the submonoids of free groups and free monoids, respectively. In this chapter we have dealt with finitely generated monoids. Some of the results exposed here fail to be true for the non-finitely generated case.