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I would like to know which is the exact difference between lattices ans Hasse diagrams. In some cases, such as when lattices are used to represent maximal and minimal ideals, it seems to me that both concepts may be empirically embodied the same way. Is that right? Which is the difference then?

Thanks in advance.

Git Gud
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Javier Arias
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    A Hasse diagram is a way to visually represent a lattice. The answer to your question is the same as the answer to the question "What is the difference between a graph (to avoid certain problems consider the axes, domain and codomain to be only integers) and its function?". – Git Gud Jul 31 '15 at 15:24
  • @GitGud Good comparison :) – rschwieb Jul 31 '15 at 15:26

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Most importantly, a lattice is a structure and a diagram is a visualization. A lattice isn't really a diagram: it is a relation with very special properties. Similarly a diagram isn't really a relation: it is more of a graph.

But of course, you can use diagrams (Hasse or otherwise) to help visualize what a lattice order is doing, or in the other direction, generate some lattice based on a diagram that you have.

A Hasse diagram is also most often used only for finite sets (but it seems harmless to call ones for infinite sets the same thing.) The most important thing to remember about Hasse diagrams is that they minimize the number of edges used. The ordering of nodes that aren't directly adjacent are stored implicitly by paths through the edges.

rschwieb
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  • Well, I guess what I meant was: Given that Hasse diagrams are used to represent lattices, 1) are there lattices which demand other visualization because Hasse diagrams do not cut it? 2) Isnt it so that Hasse diagrams are a wider object that lattices, namely, that they can represent other kinds of structures as well? If the answer to 1 and 2 is positive, examples are welcome. – Javier Arias Jul 31 '15 at 15:35
  • @JavierArias It can be problematic to do a Hasse diagram for infinite sets. You can see that here for example. You can generate a Hasse diagram for any finite poset. – rschwieb Jul 31 '15 at 15:37