What’s Abstract Algebra?

The most important branches of abstract algebra are commutative algebra, representation theory, and homological algebra. However, many other areas of mathematics have benefited from the perspective of algebra as well. Some of these include algebraic geometry, algebraic combinatorics, and even discrete mathematics. 

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What is a Group?

A group is an abstract structure containing a non-empty set and an internal binary operation that combines a pair of elements to compose a third element within the set. Some of the most famous examples of groups are circles, spheres, and conic sections. 

What Is a Ring? 

A ring is an abstract structure containing a non-empty real number set and two internal binary operations that are expressed as (R,+,). It has the characteristic that it can be divided into a smaller group called a ring. 

What Is a Body? 

A body is an abstract structure containing a non-empty complex number set and two internal binary operations that are defined as (R,+,) x (R,-). It has the characteristic that it can also be divided into smaller groups called bodies. 

What Is an Associative Algebra?

An associative algebra is an abstract structure containing a non-empty vector space and an internal binary operation that allows for the addition, subtraction, and multiplication of scalars pointwise. It has the characteristic that it can also include a bilinear associative operation, which is known as composition. 

What Is a Boolean Algebra?

A Boolean algebra is an abstract structure containing a set of xs, ys, and zs together with two constants 0 and 1. It has the characteristic that it can be a finitely axiomatizable variety. 

What Is a Magma?

Magma is an abstract structure containing a set and an internal operation that consists of a sum or a scalar product. It has the characteristic that it can be made to resemble a semigroup by using the internal operation that combines two elements to compose a third element in the set. 

What Is a Quasigroup? 

A quasigroup is an abstract structure containing a set with a single internal binary operation that is not associative. It has the characteristic that it can include a neutral element that is not a part of the set. 

What Is Unlike a Group?

An, unlike a group, is an abstract structure containing a set not composed of algebraic operations. It has the characteristic that it cannot be grouped by a scalar or an eigenvalue. 

What Is a Quasi-Monoid? 

A quasi-monoid is an abstract structure containing a set whose elements are not divisible by x, y, or z. It has the characteristic that it can not be grouped by a scalar, an eigenvalue, or a quotient of the set. 

What Is an Associative Monoid?

An associative monoid is an abstract structure that contains a set of xs, ys, and zs together with a pair of constant symbols. It has the characteristic that it can become a free commutative monoid by satisfying commutativity. 

Often the term “associative” is used to refer to a group that contains a scalar or an elliptic curve as a defining element, but in some cases, it can be applied to any other type of abstract structure. The difference is that the non-empty sets and the internal operations of an associative monoid must be commutative.