In a fully ordered group (F’,\(\leq,\circ)\), where \(\leq\) is the ordering in F’ and \(\circ\) is the group operation, we introduce an algebraic structure by inducing a further binary operation by the order and extending F’ by a zero-element \(\bar O.\) Provided with this algebraic structure, we prove in \(F^ n\), the n-fold Cartesian product of \(F:=F'\cup \{\bar O\}\), the theorems of Carthéodory and Krein-Milman. Here, Carathéodory’s theorem is proved not by a reduction step - as be usually done in linear spaces -, but by solving a certain system of equalities, which is linear with respect to the operations in F. To prove Krein-Milman’s theorem, we state some results of separation theory in such algebraic structures.