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Journal of Logic and Computation Advance Access published online on July 26, 2006
Journal of Logic and Computation, doi:10.1093/logcom/exl002
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© The Author, 2006. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org
Received October 17, 2005

Original Papers

On Weakly Cancellative Fuzzy Logics

Franco Montagna 1 *, Carles Noguera 2 *, and Rostislav Horcík 3 *

1 Department of Mathematics and Computer Science, University of Siena
2 Artificial Intelligence Research Institute (IIIA), Spanish Council for Scientific Research (CSIC)
3 Institute of Computer Science, Academy of Sciences of the Czech Republic

* To whom correspondence should be addressed.
Franco Montagna, E-mail: montagna{at}unisi.it
Carles Noguera, E-mail: cnoguera{at}iiia.csic.es
Rostislav Horcík, E-mail: horcik{at}cs.cas.cz


  Abstract

Starting from a decomposition result of monoidal t-norm-based logic (MTL)-chains as ordinal sums, we focus our attention on a particular kind of indecomposable semihoops, namely weakly cancellative semihoops. The weak cancellation property is proved to be the difference between cancellation and pseudocomplementation, so it gives a new axiomatisation of product logic and {Pi}MTL. By adding this property, some new fuzzy logics (propositional and first-order) are defined and studied obtaining some results about their (finite) strong standard completeness and other logical and algebraic properties.

Keywords: Algebraic logic; fuzzy logics; left-continuous t-norms; MTL-algebras; residuated lattices; weak cancellation.
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