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# 08: General algebraic systems

## Introduction

Here is an excerpt from the Math Reviews review of the book by Burris and Sankappanavar:
```"...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
Reviewed by R. S. Pierce
```

## Applications and related fields

"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

## Subfields

• 08B: Varieties
• 08C: Other classes of algebras

This is one of the smallest fields within the Math Reviews database.

Browse all (old) classifications for this area at the AMS.

## Textbooks, reference works, and tutorials

Some textbooks:
• "Universal algebra", by George Grätzer, Springer-Verlag, New York-Heidelberg, 1979. 581 pp. ISBN 0-387-90355-0. A good textbook.
• "Equational logic", by Walter Taylor, in Houston J. Math. 1979. This is a survey and a literature review of universal algebra.
• "The structure of finite algebras", by David Hobby and Ralph McKenzie, Contemporary Mathematics v. 76, American Mathematical Society, Providence, RI, 1988. ISBN 0-8218-5073-3
• "A course in universal algebra", by Stanley Burris and H.P. Sankappanavar, Graduate Texts in Mathematics #78, Springer-Verlag, New York-Berlin, 1981. ISBN 0-387-90578-2

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:

• "Handbook of algebra, Vol. 1" (edited by M. Hazewinkel), North-Holland Publishing Co., Amsterdam, 1996. 915 pp. ISBN 0-444-82212-7: offers a fairly comprehensive look at most algebraic structures.
• "A handbook of terms used in algebra and analysis", by A. G. Howson, Cambridge University Press, London-New York, 1972.
• "Basic structures of modern algebra", by Yuri Bahturin, Kluwer Academic Publishers Group, Dordrecht, 1993. x+419 pp. ISBN 0-7923-2459-5
• "Algebra" (2nd edition), a three-volume text by P. M. Cohn, John Wiley & Sons, Ltd., Chichester, 1991 ISBN 0-471-92840-2