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We provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators. Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order conditions on dual spaces. In particular, we specialize our general results to the variety of MV-algebras, obtaining a duality for these in which the equations axiomatizing MV-algebras are dualized as first-order conditions.
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For any MV-algebra $A$ we equip the set $I(A)$ of intervals in $A$ with pointwise L ukasiewicz negation $ eg x={ eg alphamid alphain x}$, (truncated) Minkowski sum, $xoplus y={alphaoplus betamid alpha in x,,,betain y}$, pointwise L ukasiewicz conjunction $xodot y= eg( eg xoplus eg y)$, the operators $Delta x=[min x,min x]$, $ abla x=[max x,max x]$, and distinguished constants $0=[0,0],,, 1=[1,1],,,, mathsf{i} = A$. We list a few equations satisfied by the algebra $mathcal I(A)=(I(A),0,1,mathsf{i}, eg,Delta, abla,oplus,odot)$, call IMV-algebra every model ofthese equations, and show that, conversely, every IMV-algebra is isomorphic to the IMV-algebra $mathcal I(B)$ of all intervals in some MV-algebra $B$. We show that IMV-algebras are categorically equivalent to MV-algebras, and give a representation of free IMV-algebras. We construct L ukasiewicz interval logic, with its coNP-complete consequence relation, which we prove to be complete for $mathcal I([0,1])$-valuations. For any class $mathsf{Q}$ of partially ordered algebras with operations that are monotone or antimonotone in each variable, we consider the generalization $mathcal I_{mathsf{Q}}$ of the MV-algebraic functor $mathcal I$, and give necessary and sufficient conditions for $mathcal I_{mathsf{Q}}$ to be a categorical equivalence. These conditions are satisfied, e.g., by all subquasivarieties of residuated lattices.
In their recent seminal paper published in the Annals of Pure and Applied Logic, Dubuc and Poveda call an MV-algebra A strongly semisimple if all principal quotients of A are semisimple. All boolean algebras are strongly semisimple, and so are all finitely presented MV-algebras. We show that for any 1-generator MV-algebra semisimplicity is equivalent to strong semisimplicity. Further, a semisimple 2-generator MV-algebra A is strongly semisimple if and only if its maximal spectral space m(A) does not have any rational Bouligand-Severi tangents at its rational points. In general, when A is finitely generated and m(A) has a Bouligand-Severi tangent then A is not strongly semisimple.
We prove that the category of left-handed strongly distributive skew lattices with zero and proper homomorphisms is dually equivalent to a category of sheaves over local Priestley spaces. Our result thus provides a non-commutative version of classical Priestley duality for distributive lattices and generalizes the recent development of Stone duality for skew Boolean algebras. From the point of view of skew lattices, Leech showed early on that any strongly distributive skew lattice can be embedded in the skew lattice of partial functions on some set with the operations being given by restriction and so-called override. Our duality shows that there is a canonical choice for this embedding. Conversely, from the point of view of sheaves over Boolean spaces, our results show that skew lattices correspond to Priestley orders on these spaces and that skew lattice structures are naturally appropriate in any setting involving sheaves over Priestley spaces.
We present a complete characterization of subdirectly irreducible MV-algebras with internal states (SMV-algebras). This allows us to classify subdirectly irreducible state morphism MV-algebras (SMMV-algebras) and describe single generators of the variety of SMMV-algebras, and show that we have a continuum of varieties of SMMV-algebras.
An algebra is said to be hopfian if it is not isomorphic to a proper quotient of itself. We describe several classes of hopfian and of non-hopfian unital lattice-ordered abelian groups and MV-algebras. Using Elliott classification and $K_0$-theory, we apply our results to other related structures, notably the Farey-Stern-Brocot AF C$^*$-algebra and all its primitive quotients, including the Behnke-Leptin C$^*$-algebras $mathcal A_{k,q}$.
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