A representation theorem for odd and even involutive commutative residuated chains by direct systems of abelian o-groups

Abstract: Algebraic investigations into substructural logics have been flourishing in the past decades, but the focus of this research has been fairly biased towards integral or idempotent or divisible structures which were already well-understood. On the contrary, (quasi)varieties of not necessarily integral and not necessarily divisible algebras form equivalent algebraic semantics for all the main logics in the linear and in the relevant family, including Abelian logic, and it is precisely in this area where it is possible to find very interesting connections with (lattice ordered) groups and thus with classical algebra. In this talk we address the problem of structural description of involutive commutative residuated lattices, the non-integral case. The algebras in our focus are non-divisible and non-idempotent either. Related attempts in the literature have, so far, been confined to either lattice-ordered groups (the cancellative case) or Sugihara monoids (the idempotent case). For all involutive commutative residuated chains, where either the residual complement operation leaves the unit element fixed (odd case) or the unit element is the cover of its residual complement (even case), a representation theorem will be presented in this talk by means of direct systems of abelian o-groups.

logic

Audience: researchers in the topic

Nonclassical Logic Webinar