Structure of ordered semimodules

In this paper the authors studied ordered algebraic structures (semimodules) which generalize rings, fields, modules and vector spaces as known from the theory of Algebra. Additionally these structures will be ordered and will satisfy monotonicity conditions similar to the case of ordered semigroups. In this paper we discuss the following results. (1) A linearly ordered integral domain R can be embedded in a linearly ordered field. (2) Let H be an ordered semimodule over R. (a) If H is a group then x ≤ y ⇒ x □ c ≤ y □ c for all x, y ∈ R and c ∈ H, implies x □ d ≥ y □ d for d ∈ H and d ≤ e (d ∈ H). (b) If R is a ring then a ≤ b ⇒ r □ a ≤ r □ b for all a, b and r ∈ R implies s □ a ≥ s □ b for all a, b ∈ H and s ∈R. (c) If R is the positive cone of a linearly ordered ring R and H is the positive cone of a linearly ordered group H then the external composition can be continued (extended) in a unique way on R x H such that H is a linearly ordered module over R. In fact result (1) is useful in the study of Algebraic path problems [2].