A representation theorem for finite Gödel algebras with operators

Abstract

In this paper we introduce and study finite G ¿odel algebras with operators (GAOs for short) and their dual frames. Taking into account that the category of finite Go ¿del 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 prop- erties. Our main result is a Jonsson-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.