Finite subsets R M are rational.

If R, R' are rational, then so are R R', R R' and R.

Every finitely generated submonoid N M is rational, but there also exist rational submonoids which are not finitely generated. The standard example for this is N = {(0, 0)}{ (m, n) ×|m 1 }; N is a submonoid of × but it cannot be finitely generated because for any finite set of pairs (mi , ni) the element (1, 1 + max{ni}) is in N but not in the submonoid generated by the pairs (mi , ni). The submonoid N is rational because N = {(0, 0)} {(1, 0)} + ({(1, 0)} {(0, 1)}). For groups the situation is different.

Theorem 8.21. Let H be a rational subgroup of a group G. Then H is finitely generated.

Proof: Since H is rational, there is a finite ...

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