Tensor products of modules over semifields

Abstract

A system (K, +,.) is said to be a semiField if (i) (K, +) is a commutative semigroup with identity 0 , (ii) (K\{0},.) is an abelian group with identity 1 and k. 0 = 0 . k = 0 for all k [is an element of] K , and (iii) x(y + z) = xy + xz for all x, y, z [is an element of] K.A module over a semifield K is an abelian additive group M with identity 0 , for which there is a function (k, m) -> km from K x M into M such that for all k, k[subscript 1], k[subscript 2] [is an element of] K and m, m[subscript 1], m[subscript 2] [is an element of] M , (i) k(m[subscript 1] + m[subscript 2]) = km[subscript 1] + km[subscript 2] , (ii) (k[subscript 1] + k[subscript 2])m = k[subscript 1]m + k[subscript 2]m and (iii) (k[subscript 1]k[subscript2])m = k[subscript1](k[subscript 2]m) . Moreover, if 1[subscript K]m = m for all m [is an element of] M where 1[subscript K] is the identity of (K \ {0},.) , then M is said to be a vector space over K. Let X be a subset of a vector space M over a semifield K and <X> be the subgroup of M generated by KX = {kx k | K and x[is an element of] X}. We call that X spans M if <X> = M. The set X is said to be a linearly independent set if it satisfies one of the following conditions: (i) X = [theta] or (ii) |= 1 and X [is not equal to] {0}, or (iii) |> 1 and x [is not an element of] <X\ {x}> for all x [is an element of] X. Furthermore, the set X is said to be a basis of M over K if X is a linearly independent set which spans M. In this research, we study another area in abstract algebra, a module over a semifield which is a generalization of a vector space over a field. We can define tensor products of modules over semifields and prove the Universal Mapping Property of Tensor Products. Miss Sirichan Pahupongsab studied and generalized theorems in vector spaces over fields to those in vector spaces over a semifield K such that for all [alpha], [beta] [is an element of] K there exists a [gamma] [is an element of] K which causes [alpha] + [gamma] = [beta] or [beta] + [gamma] = [alpha]. We carry on investigating and generalizing some other theorems in vector spaces over such a semifield. Besides, we can prove that every infinite basis of a vector space over a semifield has the same cardinality.