Editing Universal algebra (section)

== History ==
In [[Alfred North Whitehead]]'s book ''A Treatise on Universal Algebra,'' published in 1898, the term ''universal algebra'' had essentially the same meaning that it has today. Whitehead credits [[William Rowan Hamilton]] and [[Augustus De Morgan]] as originators of the subject matter, and [[James Joseph Sylvester]] with coining the term itself.<ref name="Gratzer.1968">{{cite book | author=George Grätzer|editor=M.H. Stone and L. Nirenberg and S.S. Chern |title=Universal Algebra|publisher=Van Nostrand Co., Inc|edition=1st| year=1968 }}</ref>{{rp|v}}

At the time structures such as [[Lie algebra]]s and [[hyperbolic quaternion]]s drew attention to the need to expand algebraic structures beyond the associatively multiplicative class. In a review [[Alexander Macfarlane]] wrote: "The main idea of the work is not unification of the several methods, nor generalization of ordinary algebra so as to include them, but rather the comparative study of their several structures."<ref>[[Alexander Macfarlane]] (1899) [https://archive.org/details/jstor-1626993 Review:''A Treatise on Universal Algebra'' (pdf)], [[Science (journal)|Science]] 9: 324–8 via [[Internet Archive]]</ref> At the time [[George Boole]]'s algebra of logic made a strong counterpoint to ordinary number algebra, so the term "universal" served to calm strained sensibilities.

Whitehead's early work sought to unify [[quaternions]] (due to Hamilton), [[Grassmann]]'s [[Exterior algebra#History|Ausdehnungslehre]], and Boole's algebra of logic.  Whitehead wrote in his book:
:''"Such algebras have an intrinsic value for separate detailed study; also they are worthy of comparative study, for the sake of the light thereby thrown on the general theory of symbolic reasoning, and on algebraic symbolism in particular. The comparative study necessarily presupposes some previous separate study, comparison being impossible without knowledge."''<ref name="Gratzer.1968"/>

Whitehead, however, had no results of a general nature. Work on the subject was minimal until the early 1930s, when [[Garrett Birkhoff]] and [[Øystein Ore]] began publishing on universal algebras.  Developments in [[metamathematics]] and [[category theory]] in the 1940s and 1950s furthered the field, particularly the work of [[Abraham Robinson]], [[Alfred Tarski]], [[Andrzej Mostowski]], and their students.<ref>Brainerd, Barron (Aug–Sep 1967) "Review of ''Universal Algebra'' by [[P. M. Cohn]]", [[American Mathematical Monthly]] 74(7): 878–880.</ref>

In the period between 1935 and 1950, most papers were written along the lines suggested by Birkhoff's papers, dealing with [[Free object|free algebras]], congruence and subalgebra lattices, and homomorphism theorems. Although the development of mathematical logic had made applications to algebra possible, they came about slowly; results published by [[Anatoly Maltsev]] in the 1940s went unnoticed because of the war. Tarski's lecture at the 1950 [[International Congress of Mathematicians]] in Cambridge ushered in a new period in which model-theoretic aspects were developed, mainly by Tarski himself, as well as C.C. Chang, [[Leon Henkin]], [[Bjarni Jónsson]], [[Roger Lyndon]], and others.

In the late 1950s, [[Edward Marczewski]]<ref>Marczewski, E. "A general scheme of the notions of independence in mathematics." Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. '''6''' (1958), 731–736.</ref> emphasized the importance of free algebras, leading to the publication of more than 50 papers on the algebraic theory of free algebras by Marczewski himself, together with [[Jan Mycielski]], Władysław Narkiewicz, Witold Nitka, J. Płonka, S. Świerczkowski, [[Kazimierz Urbanik|K. Urbanik]], and others.

Starting with [[William Lawvere]]'s thesis in 1963, techniques from category theory have become important in universal algebra.<ref>{{Citation|last1=Lawvere|first1=William F.|author-link=William Lawvere|date=1964|title=Functorial Semantics of Algebraic Theories (PhD Thesis)|url = http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html}}</ref>