Here's another perspective on Latties: It's called Formal Concept Analysis [1].
The core idea is that every binary relation (think of a binary matrix representing for example "Animal a has Property p") leads to a lattice of `concepts` associated to this relation. This lattice turns out to be very nice (it is complete: every supremum and infimum exists, even infinite ones) and conversely, every complete lattices is such a concept lattice in a very natural way (the relation that leads back to the comlete lattice is then `a is smaller than b`).
This allows to translate phenomena between the world of lattices and the world of relations with many applications in for example data mining.