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

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 ?

• @RagibZaman I'm moderately sure the OP means a lattice as in a poset with sups and infs--not a geometric lattice :) Sep 30 '13 at 5:11
• Yup, I mean a lattice as a poset with each pair of elements having LUB and GLB. Sep 30 '13 at 5:15
• @AlexYoucis Ahh of course, the mention of Hasse diagrams should have given it away. Never mind me! Sep 30 '13 at 5:15

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.