The defintion of Class of algebras

Question

Hi i'm a maths undergrad currently writing an essay on Universal algebra for my second year project.

I'm primarily using "A Course in Universal Algebra" by Burris and Sankappanavar. I can't seem to find a formal definition of class of algebras in the context of universal algebra.

Please could you provide me with a source that i could cite in a formal project that has a definition of class of algebras in the context of universal algebra. Thank you.

Answer 1

Doing a quick search through the PDF, they seem to use "class of algebras" to mean "collection of algebras we are interested in."

The use of "class" probably is to contrast with a set, since a collection of algebras often does not form a set in the technical sense.

See for example George Bergman's An Invitation to General Algebra and Universal Constructions, section 5.4:

Let us note explicitly one detail of set-theoretic language we have already used: Since all sets satisfying a given property may not together form a set, one needs a word to refer to "collections" of sets that are not necessarily themselves sets. These are called classes. An example is the class of all sets. One can think of classes which are not sets, not as actually being mathematical objects, but as providing a convenient language to use in making statements about all sets having one or another property.

Since classes are more general than sets, one may refer to any set as a "class", and this sometimes is done for reasons not involving the logical distinction, but just to vary the wording. E.g., rather than saying "the set of all those subsets of $X$ such that ..." one sometimes says "the class of those subsets of $X$ such that ...". And, for some reason, one always says "equivalence class", "conjugacy class", etc., though they are sets.

See also entries under "set vs. class" and "Axiom of Universes" in the index of that book.