Lots of people confuse relation algebra with relational algebra, including Wikipedia (until today) and one of my favorite database influencers, Jamie Brandon. I don't blame them, because the names only differ by 2 letters, and I believe not even Ted Codd knew about relation algebra; otherwise I doubt he would have named his thing "relational algebra".
If you use a database, you've probably heard of the relational algebra, which is the foundation for, well, relational databases. Codd introduced the algebra in his 1970 paper A relational model of data for large shared data banks and gave it the name "relational algebra" 2 years later. For theory fans, his seminal result (now known as Codd's theorem) says that relational algebra is equivalent to the so-called domain independent relational calculus, which is essentially a well-behaved fragment of first-order logic queries.
Far fewer people know about relation algebra outside of logic and pure math. There, it is defined as an algebraic structure abstractly by a set of axioms, and the name comes from how a concrete algebra over binary relations can be conveniently modeled by it. An interesting counterpart to Codd's theorem is that relation algebra is equivalent to \text{FOL}^3, which restricts first-order logic to use at most 3 different variables (but quantifiers can be nested arbitrarily deep). However, it can be extended to match the expressive power of FOL with a fork operator.
OK, so relational algebra is a CS thing, and relation algebra is a math thing? Not so fast! It turns out relation algebra has been quietly finding applications in computer science as well. One major piece of software built on it is the Alloy analyzer which calls it "relational logic" (which was incorrectly redirected to the page for relational algebra until I fixed it just now...). Alloy is a descendant of the Z notation by the late and great Jean-Raymond Abrial.
Let's try it again: so relational algebra is a DB thing, and relation algebra is for math and formal methods? Still not. Over the decades, there has been a small but dedicated group of people led by Dirk Van Gucht applying the ideas of relation algebra to database theory and systems. We provide some pointers to the literature in our recent paper on the Prela query language - which brings me to my final point: Prela appears to be the first query language since Van Gucht's IUGQL that is based on relation algebra.
I believe relation algebra really deserves to be known by more people. Tarski said it best: "the calculus of relations has an intrinsic charm and beauty which makes it a source of intellectual delight to all who become acquainted with it".1 Finally, I've started calling it "Tarski's Algebra of Relations", or TAR for short, in hope to avoid confusion in the future.
That's right, Tarski calls relation algebra by yet another name, "the calculus of relations", and that is again not the same as the relational calculus in database theory...↩︎