Partial metrics that are coming from polynomials

Abstract

Partial metrics were introduced to model partially defined
information, which appears in computer science. There are distance functions where the distance from a point to itself need not be equal to zero. In this paper, we introduce a class of partial metrics that is closed under addition, multiplication, supremum, and infimum. This class of partial metrics induces a topology on $\mathbb{R}$ that is finer than the upper topology. We also introduce intrinsic partial metrics and prove that these partial metrics are associated with functions derived from the supremum and infimum of polynomials. Moreover, we show that the supremum of infimum of polynomials is itself a polynomial if and only if it is equal to one of the original polynomials, and then we extend this result to rational functions.