Measures induced by units

TL;DR

This paper explores how strong units in finitely presented cancellative hoops induce automorphism-invariant measures, linking algebraic structures to measure theory and showing their mutual absolute continuity.

Contribution

It establishes a natural, representation-independent way to derive invariant measures from strong units in cancellative hoops, connecting algebraic and measure-theoretic perspectives.

Findings

Strong units induce automorphism-invariant positive linear functionals.

These functionals correspond to finite Borel measures on the spectrum.

Measures from different units are always absolutely continuous with explicit density expressions.

Abstract

The half-open real unit interval (0,1] is closed under the ordinary multiplication and its residuum. The corresponding infinite-valued propositional logic has as its equivalent algebraic semantics the equational class of cancellative hoops. Fixing a strong unit in a cancellative hoop -equivalently, in the enveloping lattice-ordered abelian group- amounts to fixing a gauge scale for falsity. In this paper we show that any strong unit in a finitely presented cancellative hoop H induces naturally (i.e., in a representation-independent way) an automorphism-invariant positive normalized linear functional on H. Since H is representable as a uniformly dense set of continuous functions on its maximal spectrum, such functionals -in this context usually called states- amount to automorphism-invariant finite Borel measures on the spectrum. Different choices for the unit may be algebraically unrelated (e.g., they may lie in different orbits under the automorphism group of H), but our second main result shows that the corresponding measures are always absolutely continuous w.r.t. each other, and provides an explicit expression for the reciprocal density.