Positive ordered 0-semifields

Abstract

A triple (K,+,.) is called a 0-semifield iff 1) (K,.) is an abelian group with zero 0, 2) (K,+) is a commutative semigroup, 3) for every x, y, z [is an element of a set] K, x(y+z) = xy + xz and 4) for every x [is an element of a set] K, x+0 = x. For a 0-semifield K, let K* denote K- {0}. A quardruple (K,+, ., [is less than or equal to]) is called a positive ordered 0-semifield if (K,+,.) is a 0-semifield and [is less than or equal to] is a partial order on K such that for every x, y, z [is an element of a set] K 1) x [is less than or equal to] y implies that x+z [is less than or equal to] y+z and xz [is less than or equal to] yz and 2) x [is more than or equal to] 0. The subset P = {x [is an element of a set] K | x [is more than or equal to] 1} of a positive ordered 0-semifield K is called the positive cone of K. Let {K[subscript i] | i [is an element of a set] I} be a family of 0-semifields. The direct product of the the family {K[subscript i] | i [is an element of a set] I} is the set of all elements (x [subscript i]) [ subscript i [is an element of a set] I ] in the cartesian product of the family {K*[subscript i | i [is an element of a set] I} and 0 where 0 = (O[subscript i]) [subscript i [is an element of a set] I] together with the componentwise operations. Let L be a subsemifield of the direct product of {K[subscript i] | i [is an element of a set] I}. L is said to be a subdirect product of {K [subscript i] | i [is an element of a set] I} iff for every j [is an element of a set] I, II [subscript j](L) = K[subscript j] where II [subscript j] is the natural projection map. A positive totally ordered 0-semifield K is said to be Archimedian iff for every x, y [is an element of a set] K*, if x<y then 1) there exists an n [is an element of a set] Z[superscript +] such that y < nx and 2) there exists an n [is an element of a set] Z such that y < x[superscript n] if x [does not equal] 1. The main results of this research are as follows : Theorem1. Let K be a 0-semifield, P [is a subset of] K* and P[superscript-1] = {x[superscript -1] | x [is an element of a set] P}. Suppose that P satisfies that 1) P is a multiplicative subsemigroup of K*, 2) P [intersection] P[superscript -1] = {1}, 3) for every x [is an element of a set] K, 1+x [is an element of a set] P and 4 ) for every x, y [is an element of a set] P and a, b [is an element of a set] K, a+b = 1 implies that ax + by [is an element of a set] P. Then there exists a unique positive compatible partial order on K induced by P such that P is the positive cone. Furthermore, there exists an order isomorphism from set of all subsets of K* which satisfy 1) - 4 onto the set of all positive compatible partial orders on K. Theorem 2. If K is a positive lattice ordered 0-semifield, then K is a subdirect product of positive totally ordered 0-semifields. Theorem 3. If K is an Archimedian positive totally ordered 0-semifield such that 1+1 [does not equal]1 and the prime semifield of K which is order isomorphic to Q[superscript+] [subscript 0] then K can be embedded into a complete positive totally ordered 0-semifield.