From Mathfuzzlog
| Authors: |
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| Title of the chapter: |
Towards the generalization of Mundici's Gamma functor to IMTL algebras: the linearly ordered case |
| Title of the book: |
Algebraic and Proof-theoretic Aspects of Non-classical Logics - Papers in honour of Daniele Mundici on the occasion of his 60th birthday |
| Editor(s): |
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| Series: |
Lecture Notes in Computer Science |
| Volume: |
4460 |
| Pages: |
{{{pages}}} |
| Publisher: |
Springer |
| City: |
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| Year: |
2008 |
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.
