A representation theorem for finite Gödel algebras with operators

 

T. Flaminio, L. Godo, R. O. Rodriguez

978-3-662-59533-6.jpgIn this paper we introduce and study finite Gödel algebras with operators (GAOs for short) and their dual frames. Taking into account that the category of finite Gödel algebras with homomorphisms is dually equivalent to the category of finite forests with order-preserving open maps, the dual relational frames of GAOs are forest frames: finite forests endowed with two binary (crisp) relations satisfying suitable properties. Our main result is a Jónsson-Tarski like representation theorem for these structures. In particular we show that every finite Gödel algebra with operators determines a unique forest frame whose set of subforests, endowed with suitably defined algebraic and modal operators, is a GAO isomorphic to the original one.

Keywords:Finite Gödel algebras; modal operators; finite forests; representation theorem.

 

In: Iemhoff R., Moortgat M., de Queiroz R. (eds). Logic, Language, Information, and Computation, WoLLIC 2019. LNCS 11541: 223–235, Springer, 2019.

Finite Gödel algebras with modal operators

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In a joint paper with Lluis Godo and Ricardo O. Rodriguez we recently studied a modal expansion of finite Gödel algebras (GAOs for short) and their dual frames. Taking into account that the category of finite Gödel algebras with homomorphisms is dually equivalent to the category of finite forests with order-preserving open maps, the dual relational frames of GAOs are forest frames: finite forests endowed with two binary (crisp) relations satisfying suitable properties. Our main result is a Jónsson-Tarski like representation theorem for these structures. In particular we show that every finite Gödel algebra with operators determines a unique forest frame whose set of subforests, endowed with suitably defined algebraic and modal operators, is a GAO isomorphic to the original one.

The paper has been published in the proceedings of WoLLIC2019 which have been held in Utrecht (Netherlands) on July 2019. Further details are available in Springer webpage: LNCS11541

IPMU 2018

 

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My contribute paper titled “Logics for strict coherence and Carnap-regular probability functions” has been recently accepted to be presented at the 17th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems – IPMU 2018.  that will take place in Cádiz, Spain.

Probability theory on product logic

Our paper (coauthored by Lluis Godo, Sara Ugolini and myself) titled “Towards a probability theory for product logic: states, integral representation and reasoning” has been accepted to be published in the International Journal of Approximate Reasoning. There, among other things, we extend probability theory from the classical to the product t-norm fuzzy logic setting. More precisely, we axiomatize a generalized notion of finitely additive probability for product logic formulas, called state, and show that every state is the Lebesgue integral with respect to a unique regular Borel probability measure.