Editing Algebraic structure (section)

== Universal algebra ==
{{main | Universal algebra}}
Algebraic structures are defined through different configurations of [[axiom]]s. [[Universal algebra]] abstractly studies such objects. One major dichotomy is between structures that are axiomatized entirely by ''identities'' and structures that are not. If all axioms defining a class of algebras are identities, then this class is a [[variety (universal algebra)|variety]] (not to be confused with [[algebraic varieties]] of [[algebraic geometry]]).

Identities are equations formulated using only the operations the structure allows, and variables that are tacitly [[universal quantifier|universally quantified]] over the relevant [[universe (mathematics)|universe]]. Identities contain no [[Logical connective|connectives]], [[Quantification (science)|existentially quantified variables]], or [[finitary relation|relations]] of any kind other than the allowed operations. The study of varieties is an important part of [[universal algebra]]. An algebraic structure in a variety may be understood as the [[quotient (universal algebra)|quotient algebra]] of term algebra (also called "absolutely [[free object|free algebra]]") divided by the equivalence relations generated by a set of identities.  So, a collection of functions with given [[signature (logic)|signatures]] generate a free algebra, the [[term algebra]] ''T''. Given a set of equational identities (the axioms), one may consider their symmetric, transitive closure ''E''.  The quotient algebra ''T''/''E'' is then the algebraic structure or variety.  Thus, for example, groups have a signature containing two operators: the multiplication operator ''m'', taking two arguments, and the inverse operator ''i'', taking one argument, and the identity element ''e'', a constant, which may be considered an operator that takes zero arguments.  Given a (countable) set of variables ''x'', ''y'', ''z'', etc. the term algebra is the collection of all possible [[term (logic)|terms]] involving ''m'', ''i'', ''e'' and the variables; so for example, ''m''(''i''(''x''), ''m''(''x'', ''m''(''y'',''e''))) would be an element of the term algebra. One of the axioms defining a group is the identity ''m''(''x'', ''i''(''x'')) = ''e''; another is ''m''(''x'',''e'') = ''x''. The axioms can be represented as [http://ncatlab.org/nlab/show/variety+of+algebras#examples_4 trees]. These equations induce [[equivalence class]]es on the free algebra; the quotient algebra then has the algebraic structure of a group.

Some structures do not form varieties, because either:

# It is necessary that 0&nbsp;≠&nbsp;1, 0 being the additive [[identity element]] and 1 being a multiplicative identity element, but this is a nonidentity;
# Structures such as fields have some axioms that hold only for nonzero members of ''S''. For an algebraic structure to be a variety, its operations must be defined for ''all'' members of ''S''; there can be no partial operations.

Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., [[Field (mathematics)|field]]s and [[division ring]]s. Structures with nonidentities present challenges varieties do not. For example, the [[direct product]] of two [[field (mathematics)|field]]s is not a field, because <math>(1,0)\cdot(0,1)=(0,0)</math>, but fields do not have [[zero divisor]]s.