Towards the generalization of Mundici's Γ functor to IMTL algebras: the linearly ordered case

  • Authors:

  • Francesc Esteva;Lluís Godo

  • Affiliations:

  • Institut d'Investigació en Intelligència Artificial, Consejo Superior de Investigaciones Científicas;Institut d'Investigació en Intel·ligència Artificial, Consejo Superior de Investigaciones Científicas

  • Venue:
  • Algebraic and proof-theoretic aspects of non-classical logics
  • Year:
  • 2007

Quantified Score

Hi-index 0.00

Visualization

Abstract

Mundici's Γ functor establishes a categorical equivalence between MV-algebras and lattice-ordered Abelian groups with a strong unit. In this short note we present a first step towards the generalization of such a relationship when we replace MV-algebras by weaker structures obtained by dropping the divisibility condition. These structures are the so-called involutive monoidal t-norm based algebras, IMTL-algebras for short. In this paper we restrict ourselves to linearly ordered IMTLalgebras, for which we show a one-to-one correspondence with a kind of ordered grupoid-like structures with a strong unit. A key feature is that the associativity property in such a new structure related to a IMTLchain is lost as soon the IMTL-chain is no longer a MV-chain and the strong unit used in Mundici's Γ functor is required here to have stronger properties. Moreover we define a functor between the category of such structures and the category of IMTL algebras that is a generalization of Mundici's functor Γ and, restricted to their linearly ordered objects, a categorical equivalence.