Duality, projectivity, and unification in ?ukasiewicz logic and MV-algebras

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• 作者：Vincenzo Marra^a ; ^{vincenzo.marra@unimi.it} ; Luca Spada^b ; ^{lspada@unisa.it}
• 关键词：primary ; 06D35 ; secondary ; 03C05 ; 57M10 ; 52B20
• 刊名：Annals of Pure and Applied Logic
• 出版年：2013
• 出版时间：March, 2013
• 年：2013
• 卷：164
• 期：3
• 页码：192-210
• 全文大小：375 K

文摘

We prove that the unification type of ?ukasiewicz (infinite-valued propositional) logic and of its equivalent algebraic semantics, the variety of MV-algebras, is nullary. The proof rests upon Ghilardi?s algebraic characterisation of unification types in terms of projective objects, recent progress by Cabrer and Mundici in the investigation of projective MV-algebras, the categorical duality between finitely presented MV-algebras and rational polyhedra, and, finally, a homotopy-theoretic argument that exploits lifts of continuous maps to the universal covering space of the circle. We discuss the background to such diverse tools. In particular, we offer a detailed proof of the duality theorem for finitely presented MV-algebras and rational polyhedra¡ªa fundamental result that, albeit known to specialists, seems to appear in print here for the first time.