Discrete dualities for n-potent MTL and BL-algebras

This talk is a contribution towards the project of developing discrete representability for the algebraic semantics of various non-classical logics. Discrete duality is a type of duality where a class of abstract relational systems is a dual counterpart to a class of algebras. These relational systems are referred to as ‘frames’ following the terminology of non-classical logics. There is no topology involved in the construction of these frames, so they may be thought of as having a discrete topology and hence the term: discrete duality. Having a discrete duality for an algebraic semantics for a logic often provides a Kripke-style semantics for the logic. In many cases it can also be used to develop filtration and tableau techniques for the logic. Another typical consequence of such a discrete duality in the case of lattice-ordered algebras is that we obtain a method of completing the algebras, i.e., an embedding of algebras into ones that are complete in the lattice sense. Establishing discrete duality involves the following steps. Given a class of algebras Alg we define a class of frames Fr. Next, for any algebra A from Alg we define its ‘canonical frame’ X (A) ∈ Fr and for each frame X in Fr we define its ‘complex algebra’ C(X) ∈ Alg. A duality between Alg and Fr holds provided that the following facts are provable: