Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality

Abstract

Abstract. We study representations of MV-algebras -equivalently, unital lattice-ordered abelian groups -through the lens of Stone-Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone-Priestley) dual spaces of MValgebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MValgebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone-Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0, 1]-valued functions on the spaces are isomorphic.

Sheaf representations and locality of Riesz spaces with order unit

“…Several sheaf representations for MV-algebras are known, we refer to Gehrke, van Gool, and Marra [24] for a complete and unifying account of them. In this section we provide two sheaf representations of RMV-algebras: Corollary 3.11 and Corollary 3.13.…”

Sheaves and duality

“…In [10], part of the results presented here were first developed to analyze sheaf representations of MV-algebra. Indeed, there, an interpolating map from X to Y ↓ was exhibited and the sheaf representation discussed above was seen to be definable from this map.…”

“…Theorem 5.7 tells us that it is precisely soft sheaf representations that are obtainable in this way. Given an MV-algebra A, the interpolating decomposition k : X → Y ↓ given in [10] may be described as follows…”

An extension of Stone Duality to fuzzy topologies and MV-algebras

“…On the other hand, the duality theory for MV-algebras boasts a rather wide interest among researchers in the area [9,11,13,14,16,18,26,28], including some of the most prominent ones, but -quite surprisingly, indeed -the only relevant work connecting MV-algebras and fuzzy topologies via a duality is, to the best of our knowledge, a paper by Maruyama [29] published in 2010. Such a circumstance is even more curious if we consider that a Stone-type representation theorem for semisimple MV-algebras was published in 1986 [2] but probably foreseen since right after the pioneering work of Chang [6].…”