Rings with Morita Duality by Weimin Xue download in ePub, pdf, iPad
Moreover, the rationale behind this list is to increase their generality and mathematical complexity. The dual cone is easier to precisely define. The purpose of this introductory list is to give examples of dualities whose structural properties appear again and again. Regarding the Boolean duality, I have to agree, I don't see a direct connection.
But it does seem to give a nice overview of the analytic parts. In particular, open and closed, interior and closure, union and intersection, are dual categories under the operation of complementation. The dual cone highlights the fact that dualities are not in all cases involutions. Further it can be shown that any functor from R-Mod to S-Mod that yields an equivalence is automatically additive.
To properly define dual polytopes you need quite some space as opposed to hand-waving where the vertices of the dual polytope lie. The dual cone is just fitting right there, I think. The introductory section is not there to present unfamiliar and interesting examples of duality, it's there to help the reader grasp what duality is. It can be shown that the left module categories R-Mod and S-Mod are equivalent if and only if the right module categories Mod-R and Mod-S are equivalent. Validity and satisfiability of formulas are dual notions and are frequently referred to as such.
The dual cone construction is unfamiliar even to many mathematicians. Both to illustrate the text and to extend it we have included a substantial number of exercises covering a wide spectrum of difficulty.
An introductory example should be something familiar to the reader, which gives the correct general idea, to be refined later. Morita equivalence takes this viewpoint to a natural conclusion by defining rings to be Morita equivalent if their module categories are equivalent. The union of an arbitrary collection of open subsets of C is an open set. This allows the definition of a functor from the category of left R-modules to the category of left Mn R -modules. For example, it is harder it seems to understand the fact that in practically all dualities you have a reversal of morphisms.
For example R possesses the least-upper-bound property. In short, the dual cone is a better example than the dual polyhedron. Every ring R has a natural R-module structure on itself where the module action is defined as the multiplication in the ring, so the approach via modules is more general and gives useful information. It would be better to have an example which most readers are likely to understand, for example the way that the octahedron is the dual of the cube. On the other hand, I would say that Boolean duality is such a basic construction of the mind that it might have been the inspiration to construct and look for all the other dualities.
Yes, for dual graph, you need a space, but not one with a metric and an origin. The text does not make it sufficiently clear that the position of the cone depends on the origin of the space. As I said before, precisely even defining the dual polyhedron requires quite some space. While it is possible to call this a duality I wonder whether people actually are doing so.
That section should not be used to teach unfamiliar material to them. There are still lots of places that need touching up, but I think the current article makes it clear that duality is a rich and interesting subject no matter how it is approached. Our general approach is categorical rather than arithmetical. Can anybody pls clean this up, since all other references to lattices in this article deal with the poset-kind of lattice.
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