Title | Towards the generalization of Mundici's Gamma functor to IMTL algebras: the linearly ordered case |

Publication Type | Book Chapter |

Year of Publication | 2007 |

Authors | Esteva F [1], Godo L [2] |

Editor | [3] |

Book Title | Algebraic and Proof-theoretic Aspects of Non-classical Logics - Papers in honour of Daniele Mundici on the occasion of his 60th birthday |

Pagination | 127-137 |

Publisher | LNAI 4460, Springer-Verlag |

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 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 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. |