Lambda the Ultimate

As a programmer with no training in type theory or abstract math (beyond what I've learned on LtU), I'm curious to know if the laws of basic, middle school, algebra can help design or understand better domain specific languages.

Recently I saw a presentation on how algebraic data types don't just contain the name 'algebra,' but actually allow many of the same operations as on simple numbers. This will be obvious to people on this forum but such ideas are very surprising to programmers at large. The link describes not just products and sums, but also x to the power of y style operations. The youtube presentation in the link above then talks about how one can even take derivatives of type expressions, which result in very useful objects, such as zippers.

There have been interesting articles[pdf] on derivatives of regular expressions. Again, kleene algebra seems to define basic operations, +, *, etc. Although the derivative part is beyond me.

Modern database systems, are based on relational algebra, and this algebra beautifully simple. Although I have't seen this algebra described specifically using basic arithmetic operators, it does contain cartesian product, sums and even division.

So many of these, very practical, tools programmers use every day are based on what looks similar to school algebra (they even contain the word algebra in their name). Where can I learn more about how to design DSLs using algebraic operators? My assumption is that by starting with a domain, and being forced to come up with interpretations of +, -, *, -, etc. (interpretations which follow specific rules), I'll end up with a pretty minimal, yet expressive, language.

Finally, I actually started thinking about this after implementing Peyton-Jones & Eber's "Composing Contracts." That language, describes a couple of basic types: Contracts and Observables. Then describes operations to on these types: And, Or, Give, Scale, which map pretty easily to +, *, negate, etc. Once again, a beautifully simple language can describe a complex set of financial instruments. What's more, the constructs of this language (like so many others), seem very closely related to school algebra! What might it mean to take the derivative of a contract? What might it mean to take it's square or square root??