Canonical extensions and relational completeness of some substructural logics * Michael Dunn Mai Gehrke † Alessandra Palmigiano ‡ March 25, 2005 Abstract In this paper we introduce canonical extensions of partially or- dered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of re- lational semantics for various substructural logics with implication as the residual(s) of fusion. 1 Introduction Canonical extensions were first introduced by J´ onsson and Tarski for Boolean algebras with operators (BAOs) in their 1950’s papers [14, 15]. Canonical extensions provide an algebraic formulation of what is otherwise treated via topological duality or relational methods. The theory of canonical exten- sions has since been simplified and generalized [9, 7, 10], leading to a widely applicable and transparent theory which is now ready to be applied even in the setting of partially ordered algebras. The only restriction is that the basic operations of the algebras to be considered either preserve or reverse the order in each coordinate. We will call such algebras monotone poset expansions (MPEs). Rather than developing a complete and general theory of canonical ex- tensions for MPEs at this stage, we have opted here to develop only what is necessary to solve a particular problem. In recent years a number of papers * The authors wish to thank an anonymous referee and M. Dunn’s student, Chunlai Zhou, for their careful reading of the manuscript and for their suggestions and corrections. † Partially supported by grant NSF01-4-21760 of the USA National Science Foundation. ‡ Partially supported by the Spanish grant MTM2004-03101 and by the grant 2001FI 00281 of the Generalitat de Catalunya. 1