Maths & Beyond

In our previous post we established a connection between skew lattices and the relational model in data science. It is thus time to answer the following question: What exactly are skew lattices? A skew lattice is a set S equipped with a pair of binary operations ∧ ( meet ) and ∨ ( join ) such that given any x, y, z in S the following identities hold: x ∧ x = x     idempotency of meet x  ∨  x  =  x     idempotency of join ( x  ∧  y ) ∧ z = x  ∧ ( y  ∧  z )    associativity of meet ( x  ∨  y ) ∨  z  =  x  ∨ ( y  ∨  z )    associativity of join x  ∧ ( x  ∨  y ) = x =  x  ∨ ( x  ∧  y )    absorption ( x  ∧  y ) ∨  y  = y = ( x  ∨  y ) ∧  y     absorption Noncommutative (in the inclusive sense as in "not necessarily commutative") lattices  were first introduced by a German physicist Pascual Jordan in 1949, however the modern theory of skew lattices is due to the extensive study developed mainly by Jonathan Leech and initiated in the paper: J. Leech, Skew lattices in rings

So, how about noncommutative lattices in data science? The aim of this post is to demonstrate a connection between data science and noncommutative lattices, more precisely skew lattices. This is work in progress. The relational model is one of the most basic models used in data science. Data is organized into tables called relations , each being composed of rows and columns. Columns correspond to attributes, while rows correspond to data entries. Rows are often referred to as  records . In data science, a database organized as a relational model can be manipulated for instance with the SQL (pronounce: "sequel") language.  Let's look at an example. The open source  Chinook database  represents a digital media store with the media related data collected from the iTunes Library. It is composed of several tables like artists, albums, tracks, genres etc. In order to get some insight into the database, let's look at the first five records of the following tables: artists