Especially in comparatively recent times, researchers in the subject of universal algebra have sought to replace that name, leading people to choose an alternative term instead. In this post, I discuss my thoughts on this matter.
Based on George Bergman’s notes, one of the main issues people identify is that the term “universal algebra” seems to suggest a singular focus on universal constructions, whereas the subject is actually more expansive in scope. However, I don’t think that this suggestion is as strong as it is implied to be, or at least strong enough to warrant a name change. “Universal” could also simply evoke universality in terms of universal applicability, which indeed is true — universal algebra is, by its very construction, applicable to all of algebra. It offers in fact a formalization of the very idea of an algebra, in the full generality of how that term can be used across all different algebras. In some sense, it is the ultimate formalization of the concept of a collection of operations on a set, and as I mentioned in my classification of the main branches of math, I would define algebra to be the study of operations. In fact, when I first heard the term “universal algebra”, I wasn’t aware of the attempts to change the name, yet I had no doubts about the “universality” of it, even before I learned about universal constructions. (I still have a lot to learn about universal constructions, UMPs, and category theory.)
The applicability of universal algebra is also universal across algebras that aren’t equational classes. It is important to note that the basic definition of an algebra given in universal algebra does not immediately make assumptions about identity axioms, but rather that equational classes are just a particularly well-behaved class of algebras (as evidenced by results like Birkhoff’s HSP theorem.) From my understanding so far, it further seems that many results in universal algebras like the Isomorphism Theorems apply to all algebras, including for example non-equational classes like fields.
Even if we do end up changing the name of universal algebra, I’d argue that the main currently suggested replacement, “general algebra,” is not satisfactory. The term “general” seems as if it could be used informally in an introduction to abstract algebra, referring simply to a generalization of an existing idea. In fact, throughout math, it is customary to hear the term “generalization” used for this purpose. Thus, assigning “general” a specific technical meaning in the context of algebra could lead to ambiguity and confusion. I’m not sure if I have a satisfactory alternate term to propose at the moment, but for now I’d at least argue in favor of the term “universal algebra” compared to “general algebra.”
