Universal algebra (general algebra)
The study not of a structure but of structures as such: what can be said about every set carrying a family of operations of stated arities, subject to stated identities, without knowing whether those operations make it a group, a ring, a lattice or a Boolean algebra. Where abstract-algebra proves a theorem once per structure, this proves it once for all structures with a given signature and reads off the group case, the ring case and the monoid case as instances.
One source: bergman-invitation-general-algebra.
The name is contested, and the book says so
george-bergman opens §1.5 with the field's own dispute. The subject "has long been called Universal Algebra, but in recent decades, many workers in the field have come to dislike this term, feeling that 'it promises too much', and/or that it suggests an emphasis on universal constructions." The favoured replacement is General Algebra, which he uses in his title while conceding it "may not be understood as referring to a specific area." Both names are live; this page keeps both.
The apparatus
- Signature and arity. An algebra is a set plus an indexed family of operations, each with an arity. Fixing the family fixes the category Ω-Alg.
- Terms. Formal expressions built from variables and the operation symbols — the syntactic layer, separate from any evaluation of them. Bergman introduces terms in Chapter 2 for groups specifically, then generalises.
- Identities and varieties. A variety is the class of all algebras satisfying a set of identities. Groups, abelian groups, rings, lattices and Boolean algebras are each a variety; fields are not, which is the standard demonstration that the definition has content.
- Free algebras. The construction that makes the whole thing work, and the reason Chapter 3 gives three separate proofs that free groups exist — from terms, as subgroups of large enough direct products, and as groups of reduced words. The prerequisite Bergman actually states is that a reader conversant with fewer than three such proofs has something to learn.
- Derived operations, clones and clonal theories. What operations a variety implicitly has beyond the ones written down, and the algebraic object that packages them.
- Structure and Semantics. Lawvere's reformulation, where a variety becomes a category and the algebras become functors out of it — the point where the subject and category-theory stop being separable.
Why it sits under the rest of the spoke
The relationship to abstract-algebra is one level of generality, not one of topic. Abstract algebra already sells the argument that proving a theorem about groups beats proving it about integers, polynomials and permutations separately. General algebra applies that argument to itself: prove it about varieties and the group theorem, the ring theorem and the lattice theorem all fall out. That is the same move made twice, and the spoke now holds both rungs.
It also brings in three subjects the spoke had no page for, because the book needs them as tools rather than as topics: order and lattice theory (semilattices, completeness, closure operators, Galois connections), set theory beyond naive use (ordinals, well-ordering, Zorn's Lemma, and a section titled "Thoughts on set theory"), and category-theory. Bergman's warning about that last one is worth carrying: for students who have not seen ordinals or categories before, the second halves of Chapters 5 and 7 are where to slow down.
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