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September 2013 Measures induced by units
Giovanni Panti, Davide Ravotti
J. Symbolic Logic 78(3): 886-910 (September 2013). DOI: 10.2178/jsl.7803100

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.

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Giovanni Panti. Davide Ravotti. "Measures induced by units." J. Symbolic Logic 78 (3) 886 - 910, September 2013. https://doi.org/10.2178/jsl.7803100

Information

Published: September 2013
First available in Project Euclid: 6 January 2014

zbMATH: 1323.03097
MathSciNet: MR3135503
Digital Object Identifier: 10.2178/jsl.7803100

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Rights: Copyright © 2013 Association for Symbolic Logic

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Vol.78 • No. 3 • September 2013
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