View source for Towards the generalization of Mundici's Gamma functor to IMTL algebras: the linearly ordered case

{{Book chapter|
author=Francesc Esteva|author2=Lluís Godo|
chapter=Towards the generalization of Mundici's Gamma functor to IMTL algebras: the linearly ordered case|
title=Algebraic and Proof-theoretic Aspects of Non-classical Logics - Papers in honour of Daniele Mundici on the occasion of his 60th birthday|
series=Lecture Notes in Computer Science|
volume=4460|
publisher=Springer|
editor=Steffano Aguzzoli| 
editor2=Agata Ciabattoni|
editor3=Brunella Gerla|
editor4=Corrado Manara|
editor5=Vincenzo Marra|
city=|
year=2008}}

== Abstract ==

Mundici's <math>\Gamma</math> 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 IMTL-algebras, 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 IMTL-chain is lost as soon the IMTL-chain is no longer a MV-chain and the strong unit used in Mundici's <math>\Gamma</math> 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 <math>\Gamma</math> and, restricted to their linearly ordered objects, a categorical equivalence.