By an \(F\)-semigroup \((S,+,\leq)\) the authors mean a commutative (additive) semigroup with an order on it satisfying the following five properties: (S1) For each \(a,b,s\in S\), \(a+s\leq b+s\) implies \(a\leq b\). (S2) If \(a\leq b\), then \(a+s\leq b+s\) for each \(s\in S\). (S3) If \(a\leq s\) and \(b\leq s\), then the supremum \(\sup\{a,b\}\), denoted by \(a\vee b\), exists. (S4) If \(s\leq a\) and \(s\leq b\), then \(\inf \{a,b\}:=a\wedge b\) exists. (S5) If for some \(a,b,s\in S\), \(a\vee b\) and \((a+s)\vee (b+s)\) exist, then \((a\vee b)+s\leq (a+s)\vee (b+s)\). That is, a commutative partially ordered semigroup, in the usual sense, which satisfies conditions (S1), (S3)--(S5). The most interesting example of an \(F\)-semigroup is the nonempty compact convex sets of a topological vector space endowed with the Minkowski sum as operation and the inclusion relation as the order. In the present paper, the authors prove that there are commutative semigroups \(S\) with an order on \(S\) satisfying the axioms (S2)--(S5) but not (S1); those satisfying (S1) and (S3)--(S5) but not (S2); those satisfying (S1), (S2), (S4) and (S5) but not (S3); those satisfying (S1)--(S3) and (S5) but not (S4); and those satisfying (S1)--(S4) but not (S5). Let now \((S,+,\leq)\) be an \(F\)-semigroup. For \((a,b)\in S\times S:=S^2\) they call the pair \((a,b)\) a fraction. They define an equivalence relation and an order relation on the set of fractions as \((a,b)\sim (c,d)\) if and only if \(a+d=b+c\), and \((a,b)\prec (c,d)\) if and only if \(a\leq c\) and \(b\leq d\), respectively. The authors define a fraction \((a,b)\) to be minimal if for any fraction \((c,d)\) such that \((c,d)\prec (a,b)\) and \((c,d)\sim (a,b)\), we have \((c,d)=(a,b)\). If \(S\) is an \(F\)-semigroup, they define the abstract difference of two elements \(a,b\in S\) as the greatest element (if it exists) of the set \(\{x\in S \mid x+b\leq a\}\) and they apply the abstract difference to characterize the minimal convex fractions.