Editing Universal algebra (section)

== Generalizations ==
{{Further|Category theory|Operad theory|Partial algebra|Model theory}}

Universal algebra has also been studied using the techniques of [[category theory]].  In this approach, instead of writing a list of operations and equations obeyed by those operations, one can describe an algebraic structure using categories of a special sort, known as [[Lawvere theory|Lawvere theories]] or more generally [[algebraic theory|algebraic theories]].  Alternatively, one can describe algebraic structures using [[monad (category theory)|monad]]s.  The two approaches are closely related, with each having their own advantages.<ref>
{{Citation|last1=Hyland|first1=Martin|last2=Power|first2=John|date=2007|title=The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads|url =http://www.dpmms.cam.ac.uk/~martin/Research/Publications/2007/hp07.pdf |archive-url=https://web.archive.org/web/20230530165901/http://www.dpmms.cam.ac.uk/~martin/Research/Publications/2007/hp07.pdf |archive-date=30 May 2023}}</ref>
In particular, every Lawvere theory gives a monad on the category of sets, while any "finitary" monad on the category of sets arises from a Lawvere theory.  However, a monad describes algebraic structures within one particular category (for example the category of sets), while algebraic theories describe structure within any of a large class of categories (namely those having finite [[product (category theory)|products]]).

A more recent development in category theory is [[operad theory]]&nbsp;– an operad is a set of operations, similar to a universal algebra, but restricted in that equations are only allowed between expressions with the variables, with no duplication or omission of variables allowed.<ref>{{Cite book |author1=Markl, M. |title=Operads in Algebra, Topology and Physics |author2=Shnider, S. |author3=Stasheff, J. D. |publisher=American Mathematical Society |year=2002 |isbn=9780821843628 |series=Mathematical Surveys and Monographs |lccn=2002016342}}</ref> Thus, rings can be described as the so-called "algebras" of some operad, but not groups, since the law {{nowrap|1=''gg''<sup>−1</sup> = 1}} duplicates the variable ''g'' on the left side and omits it on the right side.  At first this may seem to be a troublesome restriction, but the payoff is that operads have certain advantages: for example, one can hybridize the concepts of ring and vector space to obtain the concept of [[associative algebra]], but one cannot form a similar hybrid of the concepts of group and vector space.<ref>{{Cite book |author=Pierce, Richard S. |title=Associative Algebras |publisher=Springer New York |year=1982 |isbn=978-1-4757-0163-0 |location=New York, NY |pages=1–20 |chapter=The Associative Algebra |series=Graduate Texts in Mathematics |volume=88 |doi=10.1007/978-1-4757-0163-0_1}}</ref>

Another development is [[partial algebra]] where the operators can be [[partial function]]s.  Certain partial functions can also be handled by a generalization of Lawvere theories known as "essentially algebraic theories".<ref>{{nlab|id=essentially+algebraic+theory|title=Essentially algebraic theory}}</ref>

Another generalization of universal algebra is [[model theory]], which is sometimes described as "universal algebra + logic".<ref>{{cite book | isbn=0444880542 | author=C.C. Chang and H. Jerome Keisler | title=Model Theory | publisher=North Holland | series=Studies in Logic and the Foundation of Mathematics | volume=73 | edition=3rd | year=1990 |page=1}}</ref>