What is Abstract Algebra? 

Abstract algebra is the branch of mathematics that deals with the study of algebraic structures and systems. In abstract algebra, structures are classified by their number of operations, number of elements, relationship between elements and other characteristics. Typically, abstract algebra is expressed in equations. A particular class of algebras is known as a category, and its members are called objects of the category. 

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Groups are considered a basic group of abstract algebra. For example, a group is a set with a binary operation and a unary inverse. An abelian group is a set of numbers under addition and subtraction. 

Other groups that are studied in abstract algebra are rings and fields. Rings are algebraic structures that are defined on the vector space, which has a finite dimension. These structures are normally associated with Lie groups. The simplest form of a finite field is two elements, each of which is a prime. However, a finite field always has some power of the prime number. This property of a field is important in discrete mathematics. It is also used in digital electronics. 

Other structures include lattices and semilattices. Lattices are algebraic structures based on a Hasse diagram. They generalize Boolean algebras and have applications to digital electronics. Semilattices are also algebraic structures, but they are a restricted form of lattices. 

Another classification is the free C-algebras. These are sets containing linear polynomials that are closed under polynomial addition. Typical examples of free C-algebras are free monoids and commutative monoids. 

Another type of abstract algebra is the Lie algebras. Lie algebras are algebraic structures defined on the vector space. Previously, they were known as infinitesimal groups. 

A particular case of a C(A,B) is a set of all morphisms from object A to object B in a category C. If there is no proper divisor of zero, the axioms 1-9 hold. To generalize the definition, a functor F manufactures a free algebra from a field V. 

When up to isomorphism, the vector space over field F has one finite dimension. An abstract approach to the problem shows that it is a theorem. Depending on the elements in application, the law of composition is internal or external. 

There are other abstract concepts, such as the variety. Varieties are classes of sets characterized by the existence of certain patterns in their members. Examples of varieties are sets of linear polynomials, classes closed under subalgebras and quotients. 

One of the most important early developments in abstract algebra was the development of fields. In a field, the elements are a set of matrices and a set of real numbers. Typically, a field has a finite dimension, but can have more than that. Finite fields have special properties that motivated the development of abstract algebra. 

Traditionally, abstract algebra is introduced with a group or a field as a basic class. The class can be defined by listing the axioms and the unary operations involved. As a result, this method has become a standard tool for the study of algebraic structures.