Midconvex sets in Abelian groups

Анотація

A subset X of an Abelian group G is called midconvex, if for every x,y∈X the set (x+y)/2={z∈ G:2z=x+y} is a subset of X. We prove that a subset X of an Abelian group G is midconvex if and only if for every g∈G and x∈X, the set {n∈ℤ:x+ng∈X} is equal to C∩H for some order-convex set C⊆ℤ and some subgroup H⊆ℤ such that the quotient group ℤ/H has no elements of even order. This characterization implies that a subset X of a periodic Abelian group G is midconvex if and only if for every x∈X the set X-x is a subgroup of G such that every element of the quotient group G/(X-x) has odd order. Also we prove that a nonempty set X in a subgroup G⊆ℚ is midconvex if and only if X=C∩(H+x) for some order-convex set C⊆ℚ, some x∈X and some subgroup H of G such that the quotient group G/H contains no elements of even order.