Towards the generalization of Mundici's Gamma functor to IMTL algebras: the linearly ordered case
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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 | |||||
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Series: | Lecture Notes in Computer Science | |||||
Volume: | 4460 | |||||
Pages: | {{{pages}}} | |||||
Publisher: | Springer | |||||
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Year: | 2008 |
Abstract
Mundici's Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Gamma} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Gamma} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Gamma} and, restricted to their linearly ordered objects, a categorical equivalence.