Duality, projectivity, and unification in Łukasiewicz logic and MV-algebras

Abstract

We prove that the unification type of Lukasiewicz (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 -a fundamental result that, albeit known to specialists, seems to appear in print here for the first time. (Vincenzo Marra), lspada@unisa.it (Luca Spada) 1 Strictly speaking, throughout this paper we are concerned with the elementary unification type of E, meaning that in unification problems and unifiers we do not allow terms with additional function symbols not included in the signature F of E; see [3, Definition 3.9]. finitary, if V E admits no maximum, but admits finitely many maximal elements [µ 1 ], . . . , [µ u ] ∈ U (E ) ∼ such that every [σ] ∈ U (E ) ∼ lies below some [µ i ]; infinitary, if V E admits infinitely many maximal elements [µ i ] ∈ U (E ) ∼ | i ∈ I , for I an infinite index set, such that every [σ] ∈ U (E ) ∼ lies below some [µ i ];nullary, if none of the preceding cases applies.2 It would be more expedient to define unifiers for E as substitutions having a finite domain coincident with the set of variables occurring in E . This would perfectly match the definition of the pre-order V E on unifiers recalled below, which only compares unifiers on those variables occurring in E . We have nonetheless chosen to follow [3] in the basic definitions in order not to depart from established practice; cf. [3, 3.2.1] for a related discussion.3 The convention adopted in [3] is that 'τ V E σ' means 'τ is more general than σ', whereas here we are following [14] in choosing the opposite reading.

Notes on divisible MV-algebras

“…A m-simplex in [0, 1] n is the convex hull C of m + 1 affinely independent points Rational polyhedra with Z-maps form a category which is dually equivalent (see [18]) to the full subcategory of finitely presented MV-algebras. Moreover, finitely presented projective MV-algebras are in duality with retracts -by Z-maps -of finite-dimensional unit cubes.…”

“…A recent line of research in the theory of MV-algebras is the investigation of geometrical dualities, following the steps of Baker and Beynon's duality theorem between finitely presented vector spaces and polyhedra [1,2]. In [18] one can find a duality theorem for finitely presented MV-algebras and rational polyhedra with Z-maps, while in [12] finitely presented Riesz MV-algebras are proved to be dual to an appropriate category of polyhedra. In the final section of these notes we complete the framework of geometrical dualities by proving that finitely presented DMV-algebras are dual to rational polyhedra with Q-maps.…”

An analysis of the logic of Riesz spaces with strong unit

“…Let us denote by Pol R [0,1] is the category of polyhedra with R-maps as morphisms, by RatPol Q [0,1] the category of rational polyhedra with Q-maps as morphisms and by RatPol Z [0,1] the category of rational polyhedra with Z-maps as morphisms, where all polyhedra are lying in a unit cube. Following the ideas of Baker and Beynon [1,4,3], in [26,9] the authors proves that the categories RatPol Z [0,1] and MV fp are dual. This result is generalized to DMV-algebras in [21], meaning that the categories RatPol Q [0,1] and DMV fp are dual, while in [13] it is proved that Pol R [0,1] and RMV fp are dual.…”

Word problems in Elliott monoids

“…Finitely presented MV-algebras Rational polyhedra, [33] Locally finite MV-algebras Multisets, [16] MV-algebras with the map (x1, x2, . .…”