Weakly weighted generalised quasi-metric spaces and semilattices

TL;DR

This paper introduces and systematically studies weakly weighted generalised quasi-metric spaces and their connections to structures in computer science and dynamical systems, extending existing theories and exploring entropy concepts.

Contribution

It generalises the notions of weighted quasi-metrics, establishes new correspondences with other structures, and extends entropy concepts to these generalized spaces.

Findings

Established correspondences with weak partial metrics and invariant quasi-metrics.

Extended entropy notions to generalised quasi-metric semilattices.

Provided a unified framework connecting quasi-metrics, semilattices, and entropy.

Abstract

Motivated by recent applications to entropy theory in dynamical systems, we generalise notions introduced by Matthews and define weakly weighted and componentwisely weakly weighted (generalised) quasi-metrics. We then systematise and extend to full generality the correspondences between these objects and other structures arising in theoretical computer science and dynamics. In particular, we study the correspondences with weak partial metrics, and, if the underlying space is a semilattice, with invariant (generalised) quasi-metrics satisfying the descending path condition, and with strictly monotone semi(-co-)valuations. We conclude discussing, for endomorphisms of generalised quasi-metric semilattices, a generalisation of both the known intrinsic semilattice entropy and the semigroup entropy.