How to show that a lattice is isomorphic to an other lattice?

Question

By definition, a homomorphism of two lattices say g:L->S is called an isomorphism if it is one-to-one and onto. I think we can prove this by drawing Hasse diagrams for both lattice but what if the set is too big? Any analytical method ?

Answer 1

There are several methods to do that. The main question is how the lattices are given and which properties they have. In case the lattice is doubly founded, it is sufficient to consider all bijective mappings that map supremum irreducibles (such elements with one single lower neighbour) to supremum irreducibles and infimum irreducibles (with a single upper neighbour) to infimum irreducible elements. At least that is the method that is used in Formal Concept Analysis to discuss lattice properties with the use of so called reduced formal contexts. This is done in the following way:

Let $(L,≤)$ be a complete lattice, $G$ the set of supremum irreducible elements called objects, and $M$ the set of infimum irreducible elements called attributes. Then we can define a binary relation $I$ between $G$ and $M$ by the formula $I=(G×M∩{≤})$. A conclution of the fundamental theorem of Formal Concept Analysis is that two doubly founded or complete lattices are isomorphic iff their reduced contexts are isomorphic. There are additional refinements availlable in case you can fix certain automorphisms of the lattice or the context.

In case your lattice is not given as a data set, you have to use the usual algebraic methods to find a bijective mapping that is either and order isomorphism or preserves both infimum and supremum.

In case you want to get a more detailed answer, you should provide some more information about your lattice.