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

cropped-iStock-504930032-1-1.jpg

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

Hyperstates of involutive MTL-algebras and states of prelinear semihoop

The paper coauthored by Sara Ugolini and myself on hyperreal-valued probability measures (hyperstates) of involutive MTL-algebras and real-valued states of prelinear semihoop will be available soon in the proceedings of the 8th International Workshop on Logic and Cognition (WOLC2016) and it will be published by Springer (Logic in Asia).

The aim of that contribution is to provide a preliminary investigation for states of prelinear semihoops and hyperstates of algebras in the variety generated by perfect and involutive MTL-algebras (IBP0-algebras for short). Grounding on a recent result showing that IBP0- algebras can be constructed from a Boolean algebra, a prelinear semihoop and a suitably defined operator between them, our first investigation on states of prelinear semihoops will support and justify the notion of hyperstate for IBP0- algebras and will actually show that each such map can be represented by a probability measure on its Boolean skeleton, and a state on a suitably defined abelian l-group.