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Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. The term "abstract algebra" is used to distinguish the field from " elementary algebra" or "high school algebra" which teaches the correct rules for manipulating formulas and algebraic expressions involving real and complex numbers.

Historically, algebraic structures usually arose first in some other field of mathematics, were specified axiomatically, and were then studied in their own right in abstract algebra. Because of this, abstract algebra has numerous fruitful connections to all other branches of mathematics.

Examples of algebraic structures with a single binary operation are:

• semigroups
• monoids
• quasigroups
• groups

More complicated examples include:

• rings and fields
• modules and vector spaces
• associative algebraAlgebra In mathematics, an associative algebra is a vector space (or more generally module) which also allows the multiplication of vectors in a distributive and associative manner. They are thus special algebras. Definition An associative algebra A overs and Lie algebraIn mathematics, a Lie algebra (named after Sophus Lie, pronounced "lee") is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infis
• latticeSee lattice for other mathematical as well as non-mathematical meanings of the term. In mathematics, a lattice is a partially ordered set in which all nonempty finite subsets have both a supremum join and an infimum meet . On the other hand, lattices cans and Boolean algebraIn mathematics and computer science, Boolean algebras or Boolean lattices are algebraic structures which "capture the essence" of the logical operations AND, OR and NOT as well as the corresponding set theoretic operations intersection, union and complemes

In universal algebraUniversal algebra is the field of mathematics that studies the ideas common to all algebraic structures. Basic idea From the point of view of universal algebra, an algebra is a set A together with a collection of operations on A. An n- ary operation on A, all those definitions and facts are collected that apply to all algebraic structures alike. All the above classes of objects, together with the proper notion of homomorphismThis word should not be confused with homeomorphism. In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure. Some authors use the word homomorphism in a larger contex, form categories, and category theory frequently provides the formalism for translating between and comparing different algebraic structures.