Extendible Spaces

The domain theoretic notion of lifting allows one to extend a partial order in a trivial way by a minimum. In the context of Quantitative Domain Theory (e.g. [BvBR98] and [FK93]) partial orders are represented as quasi-metric spaces. For such spaces, the notion of the extension by an extremal element turns out to be non trivial. To some extent motivated by these considerations, we characterize the directed quasi-metric spaces extendible by an extremum. The class is shown to include the S-completable directed quasi-metric spaces. As an application of this result, we show that for the case of the invariant quasi-metric (semi)lattices, weightedness can be characterized by order convexity combined with the extension property. 1 Background A function d:X ×X → R+0 is a quasi-pseudo-metric iff 1) ∀x ∈ X. d(x, x) = 0 2) ∀x, y, z ∈ X. d(x, y) + d(y, z) ≥ d(x, z). A quasi-pseudo-metric space is a pair (X, d) consisting of a set X together with a quasi-pseudo-metric d on X. In case a quasi-pseudo-metric space is required to satisfy the T0-separation axiom, we refer to such a space as a quasi-metric space. In that case, condition 1) and the T0-separation axiom can be replaced by the following condition: 1′) ∀x, y. d(x, y) = d(y, x) = 0⇔ x = y. The conjugate d−1 of a quasi-pseudo-metric d is defined to be the function d−1(x, y) = d(y, x), which is again a quasi-pseudo-metric (e.g. [FL82]). The conjugate of a quasi-pseudo-metric space (X, d) is the quasi-pseudo-metric space (X, d−1). The (pseudo-)metric d∗ induced by a quasi-(pseudo-)metric d is defined by d∗(x, y) = max{d(x, y), d(y, x)}. We discuss a few examples of quasi-pseudo-metric spaces. The function d1:R →R0 , defined by d1(x, y) = y − x when x < y and d1(x, y) = 0 otherwise, and its conjugate are quasi-pseudo-metrics. We refer to d1 as the “left distance” and to its conjugate as the “right distance”. These quasi-pseudo-metrics correspond to the nonsymmetric versions of the standard metric m on the reals, where ∀x, y ∈ R.m(x, y) = |x− y|.