Theorems of convex subgroups of semifields and vector spaces over semifields

Abstract

A triple (K, +, .) is called a semifield if (1) (K, .) is an abelian group with zero 0, (2) (K, +) is a commutative semigroup with identity 0, and (3) for all x, y, z K, x(y+z) = xy+xz. A nonempty subset C={0} is a convex subgroup of K if (1) for all x, y C, y = 0 implies x/y C, and (2) for all x, y C, alpha, beta K, with alpha + beta = 1, alpha x +beta y C. A strictly finite subconvex series in K is a chain of subsemifields of K, K = K0 K1 ... Kn, such that Ki+1 is a convex subgroup of Ki and Kl = Kj for l = j. Let C and C' be two strictly finite subconvex series in K. C' is a refinement of C if every term of C appears in C'. Moreover, if C = C', then C' is a proper refinement of C. A strictly finite subconvex series in K, K = K0 K1 ... Kn {1}, is a composition series if it has no proper refinement. A vector space over a semifield K is an abelian additive group M with identity 0, for which there is a function (k, m) km from KxM into M such that for all k1, k2 K and m1, m2 M, (1) (k1k2)m1 = k1(k2m1), (2) k1(m1 + m2) = k1m1 + k1m2, (3) (k1 + k2)m1 = k1m1 + k1m1 and (4) 1Mm1 = m1. Let B be a subset of a vector space M over K and is the subgroup of M generated by KB = {kb / k K and b B}. We call that B spans M if = M. A set B is said to be a linearly independent set if it satisfies one of the following conditions: (1) B = phi, or (2) /B/ = 1 and B = {0), or (3) /B/ > 1 and b for all b B. A set B is said to be a basis of a vector space M over K if B is a linearly independent set which spans M and we say that M is finite-dimensional if M has a finite basis. The main results of this research are follows: Theorem Let K be a semifield which has a composition series. Then any two composition series are equivalent. Theorem Let A and B be finite subsets of a vector space M over a semifield K which satisfies the property (*), i.e., for all alpha, beta K there exists a gamma K such that alpha + gamma = beta or beta + gamma = alpha. If they are bases of M, then /A/ = /B/. Zassenhaus Lemma, Schreier's Theorem and standard theorems in vector spaces over a field can be extended in vector spaces over a semifield which satisfies the property (*).