| Universal Algebra |
Article Index for Universal |
Website Links For Universal |
Information AboutUniversal Algebra |
| CATEGORIES ABOUT UNIVERSAL ALGEBRA | |
| abstract algebra | |
| universal algebra | |
| algebraabstract algebra | |
| universal algebra | |
| algebra | |
| abstract algebra | |
| category theory | |
|
BASIC IDEA
Universal algebra can be seen as a special branch of Model Theory , in which we are dealing with structures having operations only (i.e., no Relations except for equality), and in which the language used to talk about these structures uses equations only. On the other hand the structures are such that they can be defined in any Category which has ''finite Product s''. EXAMPLES Most of the usual algebraic systems of mathematics are examples of universal algebras, but not always in an obvious way. Groups
(Of course, we usually write "''x ''-1" instead of "~''x''", which shows that the notation for operations of low Arity is not ''always'' as given in the second paragraph.) It's important to check that this really does capture the definition of a group. The reason that it might not is that specifying one of these universal groups might give more information than specifying one of the usual kind of group. After all, nothing in the usual definition said that the identity element ''e'' was ''unique''; if there is another identity element ''e''', then it's ambiguous which one should be the value of the nullary operator ''e''. However, this is not a problem, because Identity Element s can be proved to be always unique. The same thing is true of Inverse Element s. So the universal algebraist's definition of a group really is equivalent to the usual definition. BASIC CONSTRUCTIONS We assume that the type, , has been fixed. Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product. A . In particular, we can take the homomorphic image of an algebra, ''h''(''A''). A subalgebra of ''A'' is a subset of ''A'' that is closed under all the operations of ''A''. A product of some set of algebraic structures is the Cross Product of the sets with the operations defined coordinatewise. SOME BASIC THEOREMS
MOTIVATIONS AND APPLICATIONS In addition its unifying approach, Universal algebra also gives deep theorems and important examples and counterexamples. It provides a useful framework for those who intend to start the study new classes of algebras. It can enable the use of methods invented for some particular classes of algebras to other classes of algebras, by recasting the method in terms of universal algebra (if possible), and then interpreting these as applied to other classes. It has also provided conceptual clarification; as J.D. H. Smith puts it, ''"What looks messy and complicated in a particular framework may turn out to be simple and obvious in the proper general one."'' In particular, universal algebra can be applied to the study of Monoid s, Rings , and Lattice s. Before universal algebra came along, many theorems (most notably the Isomorphism Theorem s) were proved separately in all of these fields, but with universal algebra, they can be proven once and for all for every kind of algebraic system. In his 1963 thesisF. William Lawvere, Functorial Semantics of Algebraic Theories, Dissertation, Columbia University, 1963. http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html, William Lawvere showed that every algebraic theory corresponds to a cartesian category (i.e. given objects ''X'' and ''Y'', there is a product object ''X × Y''), and conversely, every cartesian category can be expressed in terms of an algebraic theory. A functor from the category into the category of sets assigns a set to each type and a function to each function symbol satisfying the axioms. Every group, for instance, arises as a functor F:Th('''Grp''')→ '''Set''', where Th('''Grp'''), is the theory of groups described above. FURTHER ISSUES A more generalised program along these lines is carried out by Category Theory . Given a list of operations and axioms in universal algebra, the corresponding algebras and homomorphisms are the objects and morphisms of a Category . Category theory applies to many situations where universal algebra does not, extending the reach of the theorems. Conversely, some theorems that hold in universal algebra do not generalise all the way to category theory. Thus both fields of study are useful. 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 itselfGrätzer, George. Universal Algebra, Van Nostrand Co., Inc., 1968, p. ''v''.. At the time structures such as '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 Calculus Of Extensions , 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." 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 (Brainerd 1967). In the period between 1935 and 1950, most papers were written along the lines suggested by Birkhoff's papers, dealing with 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 , R. C. Lyndon, and others. In the late 1950s, E. MarczewskiMarczewski, E. "A general scheme of the notions of independence in mathematics." Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 6 (1958), 731-736. 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 J. Mycielski, W. Narkiewicz, W. Nitka, J. Płonka, S. Świerczkowski, K. Urbanik, and others. SEE ALSO FOOTNOTES REFERENCES
|
|
|