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Measures Induced by Units

Published online by Cambridge University Press:  12 March 2014

Giovanni Panti  and
Davide Ravotti
Giovanni Panti
Affiliation:
Department of Mathematics, University of Udine, Via Delle Scienze 206, 33100 Udine, Italy, E-mail: giovanni.panti@uniud.it
Davide Ravotti
Affiliation:
Department of Mathematics, University of Udine, Via Delle Scienze 206, 33100 Udine, Italy, E-mail: davide.ravotti@gmail.com
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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.

Keywords

03G2506F2011K06cancellative hooplattice-ordered abelian groupMV-algebrastrong unituniform distributionCesàro mean
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 3 , September 2013 , pp. 886 - 910
Copyright
Copyright © Association for Symbolic Logic 2013

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