Three-polar space over the semi-field of double numbers

Abstract

Multi-polarity is a generalization of the term of vector space, where a semi-field is used instead of a field. In this paper, we deal with a three-polar space over a semi-field of double numbers. We also introduce operations of addition and multiplication such that they form a commutative ring with unit on the three-polar space. Both these operations are isomorphic to some operations on the set $\mathbb{R}^4$.