Criteria for existence of semigroup homomorphisms and projective rank functions
نویسنده
چکیده
Suppose A, S, and T are semigroups, e: A → S and f : A → T semigroup homomorphisms, and X a generating set for S. We assume (1) that every element of S divides some element of e(A), (2) that T is cancellative, (3) that T is power-cancellative (i.e, xd = yd ⇒ x = y for d > 0), and (4) a further technical condition, which in particular holds if T admits a semigroup ordering with the order-type of the natural numbers. We show that there then exists a homomorphism S → T making a commuting triangle with e and f if and only if for every relation w(x1 , ... , xn ) = e(a) holding in S, with x1 , ... , xn ∈X, a ∈A, and w a semigroup word, there exist t1 , ... , tn ∈T satisfying w(t1 , ... , tn ) = f (a). This leads to an arithmetic criterion for the existence of integer-valued projective rank functions on rings.
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