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.
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.