Polyadic De Morgan Monoids: Functional representation and MG structures

Abstract

In this paper we explore the class of polyadic De Morgan Monoids (DMMs). We show that we can define functional polyadic DMMs such that, drawing on the work of Pigozzi and Salibra on polyadic non-classical logic and abstract variable binding calculi, we can prove a functional representation theorem for polyadic DMMs. DMMs are the algebraic semantics of the relevant logic R. We show a similar correspondence between polyadic DMMs and the first-order relevant logic RQ. This is done by showing that we can transform functional polyadic DMMs into equivalent MG structures and conversely, where MG structures are known to be a semantics for RQ. Indeed, we show the sense in which it can be claimed that MG structures contain equivalent functional polyadic DMMs. The main results of the paper are (i) a functional representation theorem, and (ii) the equivalence of MG structures with functional polyadic DMMs.