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BASIC IDEA


  •  ''y''. Operations of higher or unspecified arity are usually denoted by function symbols, with the arguments placed in parentheses and separated by commas, like ''f''(''x'',''y'',''z'') or ''f''(''x''1,...,''x''''n''). Some researchers allow Infinitary operations, such as \bigwedge_{\alpha} x_\alpha, allowing the algebraic theory of Complete Lattice s to be studied.


  •  (''y'' --- ''z'') = (''x'' --- ''y'') --- ''z''. The axiom is intended to hold for all elements ''x'', ''y'', and ''z'' of the set ''A''.


According to Yde Venema, "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 ), and in which the language we use 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



Groups


  • , subject to these axioms:

  • Associativity (as in the previous paragraph): ''x'' --- (''y'' --- ''z'')  =  (''x'' --- ''y'') --- ''z''.

  • Identity Element : There exists an element ''e'' such that ''e'' --- ''x''  =  ''x''  =  ''x'' --- ''e''.

  • Inverse Element : For each ''x'', there exists an element ''i'' such that ''x'' --- ''i''  =  ''e''  =  ''i'' --- ''x''.

  •  ''y'' belongs to the set ''A'' whenever ''x'' and ''y'' do. But from a universal algebraist's point of view, that is already implied when you call --- a binary operation.)


  • , then list the axioms as follows:

  • Associativity: ''x'' --- (''y'' --- ''z'')  =  (''x'' --- ''y'') --- ''z''.

  • Identity element: ''e'' --- ''x''  =  ''x''  =  ''x'' --- ''e''.

  • Inverse element: ''x'' --- (~''x'')  =  ''e''  =  (~''x'') --- ''x''.

    • (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.)


Now, 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 require more information than specifying one of the usual kind of groups. After all, nothing in the definition of group said that the identity element ''e'' was ''unique''; if there is another identity element ''e''', then it's ambiguous as to which should be the value of the nullary operator ''e''. However, this is not a problem, because Identity Element s are always unique. The same thing is true of Inverse Element s. So the universal algebraist's definition of group really is equivalent to the usual definition.


FURTHER ISSUES


  • is a binary operation, then ''h''(''x'' --- ''y'') = ''h''(''x'') --- ''h''(''y''). And so on. A few of the things that can be done with homomorphisms, as well as definitions of certain special kinds of homomorphisms, are listed under the entry Homomorphism .


This article is too brief to indicate the breadth of the results of universal algebra. The motivation for the field is the many examples of algebras (in the sense of universal algebra), such as 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, you can prove them once and for all for every kind of algebraic system.

A more generalised program along these lines is carried out by Category Theory . 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 just don't generalise all the way to category theory. Thus both fields of study are useful. The connection is that given a list of operations and axioms, the corresponding algebras and homomorphisms are the objects and morphisms of a Category .


SEE ALSO



REFERENCES