Partial metrics and normed inverse semigroups (Q6582348)

This paper presents an attempt to unify ways of assigning norms and\slash or distances to disparate structures. The author considers inverse semigroups with identity (inverse monoids), where the `inverse' refers to the existence, for every~\(x\), of a unique element \(x^*\) such that \(x+x^*+x=x\) and \(x^*+x+x^*=x^*\). The basic notion is that of a submodular function: \(p:X\times X\to\mathbb{R}\cup\{-\infty\}\) is submodular if \(p(x,z)+p(y,y)\le p(x,y)+p(y,z)\) for all \(x\), \(y\), and~\(z\); clearly pseudo-metric are special cases of such functions. After a short investigation this is specialized to partial pseudo-metrics, which are symmetric submodular functions that additionally satisfy \(0\le p(x,x)\le p(x,y)\) for all~\(x\) and~\(y\). These partial pseudo-metrics have in turn `intrinsic' pseudo-metric associated with them, obtained by subtracting \(\min\{p(x,x),p(y,y)\}\) or \(\frac12(p(x,x)+p(y,y))\) from \(p(x,y)\) respectively, and a third one defined by \(\sqrt{p(x,y)^2-p(x,x)\cdot p(y,y)}\). These are all equivalent in that they generate the same topology; the first two are even Lipschitz-equivalent.\N\NThis all leads to a theorem stating that, given an inverse monoid \((S,+,0)\) that satisfies \(x+x^*=x^*+x\) and that has a semigroup norm \(\|{\cdot}\|\) one can define a right-invariant (pseudo-)metric on \(S\) that interacts well with the intrinsic order-theoretic and algebraic structure of~\(S\). There are no explicit examples, but the author shows how this results apply to various kinds of structures.