Description
An algebraic counterpart of Abelian Logic is the class of linearly ordered abelian groups. An algebraic counterpart of IUML*, which is a logic at the intersection of relevance logic and many-valued logic, is the class of odd Sugihara monoids. These classes of algebras are both particular instances of odd involutive residuated chains, and seem to be far from each other, suggested by the fact that there is a single idempotent element in linearly ordered abelian groups, whereas all elements are idempotent in odd Sugihara monoids.
We shall present a representation theorem for odd (and also for even) involutive residuated chains by means of linearly ordered abelian groups and a quite difficult construction, thus showing that all odd (and even) involutive residuated chains can be obtained in a uniform manner from quite specific ‘building blocks’. If time permits we shall treat separately the finitely generated case by employing a different approach. Finally, we shall demonstrate how the representation theorem can be applied to obtain a completeness result of IUL$^{fp}$ the Involutive Uninorm Logic with Fixed Point, the latter result being a first completeness result of a substructural logic lacking the weakening rule.