An algebraic ordered extension of vector space

Abstract

In this paper we have discussed an algebraic ordered extension of vector space. This new structure comprises a semigroup structure, a scalar multiplication and a compatible partial order. It is an algebraic axiomatisation of topological hyperspace; also it can be thought of as a generalisation of vector space in the sense that, it always contains a vector space and conversely, every vector space can be embedded maximally into such a structure. Initially the idea of this structure was given by S. Ganguly et al. with the name “quasi-vector space” in “An Associated Structure Of A Topological Vector Space; Bull. Cal. Math. Soc; Vol-96, No.6 (2004), 489-498”. The axioms of this structure evolve a very rapid growth of its elements with respect to the partial order and also evoke some sort of positiveness in each element. Meanwhile, a vector space is evolved within this structure and positivity of each element of the new structure is judged with respect to the elements of the vector space generated. Considering the exponential behaviour of its elements, we have studied this structure in the present paper with a new nomenclature —“exponential vector space” in short ‘evs’. We have developed a quotient structure on an evs by defining ‘congruence’ on it and have shown that the quotient structure also forms an evs with respect to suitably defined operations and partial order. We have obtained an isomorphism theorem and a correspondence theorem in the context of exponential vector space. Further, we have topologised the quotient evs by defining compatibility of the associated congruence with the topology of the base evs. A necessary and sufficient condition has been deduced so that the order-isomorphism stated under the isomorphism theorem becomes topological. Also, we have constructed order-morphisms on a quotient evs corresponding to that on the base evs.