08: General algebraic systems
"...Algebra has had a long association with universality. Newton's lectures on algebra were published in 1707 as Arithmetica universalis. However, the current meaning of the expression "universal algebra" dates from the work of Birkhoff and Ore in the 1930s. The appeal of the subject in its early years was probably due to its universality, but the work of a few dozen people during the past two decades has added a dimension of depth to the breadth that was the original trademark of universal algebra. " Reviewed by R. S. Pierce © Copyright American Mathematical Society 1983, 1997
For more information about this field, see that review (83k:08001) or 94d:08001.
"Algebra" is a very broad section of mathematics; there are separate index pages here for specific algebraic categories (groups, fields, etc.) This heading focuses both on the broad principles covering all of algebra and on specific algebraic constructs not included in those other areas. By extension (and somewhat inappropriately) we use it to house a few resources discussing many areas of algebra.
Universal algebra is arguably more a topic in Logic (03C05) (Model Theory), hence there is significant overlap.
For Boolean algebras and generalizations see Ordered algebraic structures (06E).
For groupoids, semigroups, and other multiplicative sets see Group Theory (sections 20L, 20M, 20N).
There is a Ring FAQ delineating some of the field-like structures such as division rings.
"Varieties" in this sense have nothing to do with varieties in Algebraic Geometry
This is one of the smallest fields within the Math Reviews database.
Browse all (old) classifications for this area at the AMS.
There is a review volume, surveying much of the literature through 1988: Consult Math Reviews (review 91c:08001) for details. (The survey is in Russian and not readily available to me.) See also Featured Review MR97e:08002 (by Joel Berman) of some papers by Ralph McKenzie for a further overview of recent results in finite algebras and equational logic.
This is perhaps the most appropriate page to list some texts and resources applicable to many areas of abstract algebra: