## Journal of Symbolic Logic

### Measures induced by units

#### 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.

#### Article information

Source
J. Symbolic Logic, Volume 78, Issue 3 (2013), 886-910.

Dates
First available in Project Euclid: 6 January 2014

https://projecteuclid.org/euclid.jsl/1389032280

Digital Object Identifier
doi:10.2178/jsl.7803100

Mathematical Reviews number (MathSciNet)
MR3135503

Zentralblatt MATH identifier
1323.03097

Subjects
Primary: 03G25; 06F20; 11K06.

#### Citation

Panti, Giovanni; Ravotti, Davide. Measures induced by units. J. Symbolic Logic 78 (2013), no. 3, 886--910. doi:10.2178/jsl.7803100. https://projecteuclid.org/euclid.jsl/1389032280