2 ms·

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 represent

by Garlef 6y ago

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. [1] https://en.wikipedia.org/wiki/Formal_concept_analysis https://en.wikipedia.org/wiki/Formal_concept_analysis

tom_mellior 6y ago
This looks interesting! I hadn't heard of it before. But I do think that saying that "every supremum and infimum exists" is overselling it a bit. They exist, but from the example in the Wikipedia article it looks like they only exist trivially because you define unique supremum and infimum nodes. In the body of water classification example on Wikipedia the bottom-most node is a "concept" of a body of water that is both "temporary" and "constant" at the same time. As these are antonyms, this concept is unsatisfiable, and the node is entirely artificial, not corresponding to any possibly existing object. I haven't tried to understand the more advanced extensions listed on the Wikipedia page, they might be able to rectify this kind of problem.
danharaj 6y ago
The rabbit hole goes deep here. Formal concept analysis is essentially the study of certain galois connections on lattices derived from relations on sets. You can generalize relations to profunctors on categories and galois connections to adjunctions. https://link.springer.com/chapter/10.1007%2F978-3-642-29892-9_24 https://link.springer.com/chapter/10.1007%2F978-3-642-29892-... https://golem.ph.utexas.edu/category/2013/08/the_nucleus_of_a_profunctor_so.html https://golem.ph.utexas.edu/category/2013/08/the_nucleus_of_... Tai-Danae Bradley works with these ideas applied to probability distributions on a product space. The same formal ideas developed in this linear algebraic context lead to a very rich theory. https://johncarlosbaez.wordpress.com/2020/05/07/formal-concepts/ https://johncarlosbaez.wordpress.com/2020/05/07/formal-conce...

by Garlef 6y ago

tom_mellior 6y ago

This looks interesting! I hadn't heard of it before. But I do think that saying that "every supremum and infimum exists" is overselling it a bit. They exist, but from the example in the Wikipedia article it looks like they only exist trivially because you define unique supremum and infimum nodes. In the body of water classification example on Wikipedia the bottom-most node is a "concept" of a body of water that is both "temporary" and "constant" at the same time. As these are antonyms, this concept is unsatisfiable, and the node is entirely artificial, not corresponding to any possibly existing object. I haven't tried to understand the more advanced extensions listed on the Wikipedia page, they might be able to rectify this kind of problem.

danharaj 6y ago

The rabbit hole goes deep here. Formal concept analysis is essentially the study of certain galois connections on lattices derived from relations on sets. You can generalize relations to profunctors on categories and galois connections to adjunctions. https://link.springer.com/chapter/10.1007%2F978-3-642-29892-9_24 https://link.springer.com/chapter/10.1007%2F978-3-642-29892-... https://golem.ph.utexas.edu/category/2013/08/the_nucleus_of_a_profunctor_so.html https://golem.ph.utexas.edu/category/2013/08/the_nucleus_of_... Tai-Danae Bradley works with these ideas applied to probability distributions on a product space. The same formal ideas developed in this linear algebraic context lead to a very rich theory. https://johncarlosbaez.wordpress.com/2020/05/07/formal-concepts/ https://johncarlosbaez.wordpress.com/2020/05/07/formal-conce...