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Convex polytopes and retractions of Abelian groups

Introduction. For any group C7, let F(G) denote the semigroup of finite non-empty subsets of G. A semigroup homomorphism a: F(G) -> G satisfying a({g}) = g for all g in G is called a retraction of G. The notion of a group admitting a retraction generalizes the notion of a latticeordered group because in any lattice-ordered group the mapping A -*• A A is a retraction (cf., [1]). This example of a retraction induced by a lattice order has the property that the effect of the mapping on F(G) is determined uniquely by its effect on two element subsets. This is not so for all retractions, and [1, example 6.1], gives an instance where two distinct retractions agree on all two element subsets. The question of whether distinct retractions can agree on sets of cardinality less than or equal to n for arbitrary n is dealt with in this paper. Also, in looking at a retraction a on a group G, the notion which corresponds to that of an 1-subgroup is the notion of a ^-subgroup—a subgroup H of G such that a restricted to F(G) is a retraction of H. In this paper we also deal with the question of whether a subgroup H of G with the property that all sets in F(H) of cardinality less than n get mapped by a io H must necessarily be a er-subgroup. Our approach considers only retractions of divisible abelian groups and builds on observations made in [3] and [4]. In the process of studying retractions we get a correspondence between retractions and homomorphisms from a semigroup of convex polytopes in Q to Q, so some of our results are essentially geometric in nature.