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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.
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We characterize the left-handed noncommutative frames that arise from sheaves on topological spaces. Further, we show that a general left-handed noncommutative frame $A$ arises from a sheaf on the dissolution locale associated to the commutative shadow of $A$. Both constructions are made precise in terms of dual equivalences of categories, similar to the duality result for strongly distributive skew lattices in arXiv:1206.5848.
We extend Stone duality between generalized Boolean algebras and Boolean spaces, which are the zero-dimensional locally-compact Hausdorff spaces, to a non-commutative setting. We first show that the category of right-handed skew Boolean algebras with intersections is dual to the category of surjective etale maps between Boolean spaces. We then extend the duality to skew Boolean algebras with intersections, and consider several variations in which the morphisms are restricted. Finally, we use the duality to construct a right-handed skew Boolean algebra without a lattice section.
For an efficient implementation of Buchbergers Algorithm, it is essential to avoid the treatment of as many unnecessary critical pairs or obstructions as possible. In the case of the commutative polynomial ring, this is achieved by the Gebauer-Moeller criteria. Here we present an adaptation of the Gebauer-Moeller criteria for non-commutative polynomial rings, i.e. for free associative algebras over fields. The essential idea is to detect unnecessary obstructions using other obstructions with or without overlap. Experiments show that the new criteria are able to detect almost all unnecessary obstructions during the execution of Buchbergers procedure.
Let $A$ be an algebra over a commutative ring $R$. If $R$ is noetherian and $A^circ$ is pure in $R^A$, then the categories of rational left $A$-modules and right $A^circ$-comodules are isomorphic. In the Hopf algebra case, we can also strengthen the Blattner-Montgomery duality theorem. Finally, we give sufficient conditions to get the purity of $A^circ$ in $R^A$.
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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