Timeline for Was lattice theory central to mid-20th century mathematics?
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7 events
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| Jul 4, 2014 at 10:05 | comment | added | Gejza Jenča | @JosephVanName I agree with everything you wrote. However, if we define "lattice theory" as "what lattice theorists did and still do", I think that wast majority of the work in this area concerns $\mathbf{Lat}$ -- sublattices, congruences, extensions etc. etc. I do not see many papers on other types of maps between lattices, the ones you (rightly so!) pointed out as "more natural to consider". I admit I was a little bit oversimplifying when I put just $\mathbf{Pos}$ into the codomain of the functor. I should use $\mathbf{Sem}$ instead. | |
| Jun 25, 2014 at 7:26 | comment | added | Joseph Van Name | I should mention that I do not believe that it is a disadvantage to consider maps that only preserve joins and meets between complete lattices for two reasons. First, if every subset of a poset has a least upper bound, then that poset is automatically complete lattice. Therefore, it would seem in a sense more natural to consider the category $\mathbf{S}$ mentioned above as opposed to the category of complete lattices. Secondly, the mappings between complete lattices that preserve arbitrary joins are precisely the left Galois adjoints of the maps that preserve arbitrary meets. | |
| Jun 25, 2014 at 7:18 | comment | added | Joseph Van Name | For example, if $H$ is a group, then $F,G$ could be the functors where $F(H)=G(H)$ is the lattice of all subgroups of $H$ where if $f:H_{1}\rightarrow H_{2}$ is a group homomorphism, then $F(f)(K)=f[K]$ and $G(f)(K)=f^{-1}[K]$. | |
| Jun 25, 2014 at 7:14 | comment | added | Joseph Van Name | You said that we only have a functor $F:\mathbf{C}\rightarrow\mathbf{Pos}$, but it turns out that one usually obtains a covariant functor $F:\mathbf{C}\rightarrow\mathbf{S}$ were the objects in $\mathbf{S}$ are the complete lattices and the morphisms between objects in $\mathbf{S}$ are the functions preserving arbitrary joins. Alternatively, one usually also obtains a contravariant functor $G:\mathbf{C}\rightarrow\mathbf{T}$ where the objects in $\mathbf{T}$ are the complete lattices and the morphisms are the mappings preserving arbitrary meets. | |
| Jun 25, 2014 at 7:04 | comment | added | Joseph Van Name | It turns out that the category of matroids (where the morphisms are the maps where the inverse image of a flat is a flat) is contravariantly equivalent to the category of all finite atomistic semimodular lattices where the morphisms are not lattice homomorphisms but join-preserving mappings that map atoms to atoms. It is often the case that one wants the mappings between certain lattices to preserve just joins or something like that. For example, in point-free topology the frame homomorphisms preserve arbitrary joins but only finite meets. | |
| Feb 3, 2014 at 10:37 | history | edited | Gejza Jenča | CC BY-SA 3.0 |
grammar fixed
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| Feb 3, 2014 at 10:30 | history | answered | Gejza Jenča | CC BY-SA 3.0 |