A Representation Theorem for Finite Gödel Algebras with Operators
In 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.
KeywordsFinite Gödel algebras Modal operators Finite forests Representation theorem
The authors acknowledge partial support by the SYSMICS project (EU H2020-MSCA-RISE-2015 Project 689176). Further, Flaminio acknowledges partial support by the Spanish Ramon y Cajal research program RYC-2016-19799; Flaminio and Godo by the Spanish FEDER/MINECO project TIN2015- 71799-C2-1-P; Rodriguez, by the projects UBA-CyT: 20020150100002BA and PIP 112-2015-0100412 CO.
- 1.Aguzzoli, S., Bova, S., Gerla, B.: Free algebras and functional representation for fuzzy logics. In: Cintula, P., Hájek, P., Noguera, C. (eds.) Handbook of Mathematical Fuzzy Logic - Volume 2, Chap. IX. Studies in Logic, vol. 38, pp. 713–791. College Publications, London (2011)Google Scholar