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What is abstract algebra?

Abstract algebra is the branch of algebra dealing with the study of algebraic systems or structures with one or more mathematical operations associated with elements with an identifiable pattern, differing from the usual number systems. In abstract algebra, the elements combined to perform mathematical operations are not interpretable as numbers, hence its abstract nature.

The elements of abstract algebra operate as an abstraction of the algebraic properties common to different number systems and other objects of mathematical study. Therefore, its objective is to learn about the properties of operations, regardless of the operands’ characteristics. Most of this branch was created in the 19th century to respond to the need for greater accuracy in mathematical definitions.

What can abstract algebra be used for?

The main purpose of abstract algebra is analyzing a set endowed with one or more operations with special characteristics or properties to learn about the relationships between those properties of the operations in a precise way, as well as the consequences and possible results of their associations.

What are algebraic structures?

In abstract algebra, “structure” means a set or group with one or more algebraic operations. These structures are classified according to the number of operations that may exist in them, as well as by their characteristics, the number of elements or subsets, and the relationship that exists among the elements of the main set, regardless of their nature.

What is the Law of Composition?

This is a term from abstract algebra used to name a type of binary operation where two elements of given sets are assigned to another element, giving rise to distinct algebraic structures. The law of composition can be internal or external, depending on whether the elements in the application are part of the same set or of different sets, respectively.

The internal laws of composition are represented by the following symbols: ⊙, ⊚, ⊕, ⊖, ⊛, ⊗, and ⊘. The external laws of composition are represented by the following symbols: ⋅, ∘, +, -, ⋇.x,  and ∕. The sets are represented by capital letters (A, B, C…) and their elements with lowercase letters (a, b, c…).

What types of algebraic structures are there?

The most common algebraic structures from the abstract branch can be classified as:

  1. A single law of composition or binary operation:
    • Magmas: algebraic structures of the form (A,⊚) where A is a set with a single internal binary operation.
    • Semigroups: structures of the form (A,⊚) where A is a non-empty set and ⊚ is an internal operation defined in A.
    • Groups: algebraic structures formed by a non-empty set with an internal operation that combines a pair of elements to compose a third element within a set.
    • Quasigroups: algebraic structures with a linear term that are configured like a magma with a single law of internal composition whose elements are divisible. Their main difference from groups is that they are not necessarily associative.  
    • Monoids: algebraic structures with associative operations and a neutral element; the latter is what differentiates them from semigroups.
  2. Has two or more laws of composition or binary operations:
    • Rings: algebraic systems composed of a set and two internal binary operations that are expressed as (R,+,∙).
    • Bodies (also called fields): they are commutative rings of division.
    • Modules: algebraic structures that act in group representation theory, where a group entails concrete transformations of a mathematical object.
    • Vector spaces (also called linear spaces): algebraic structures composed of a non-empty set, an internal operation (called a sum), and an external operation (called a scalar product). The elements of a vector space are called vectors.
    • Associative algebras: they are modules that also allow the multiplication of vectors in a distributive and associative way.
    • Lie algebras: algebraic structures defined on a vector space and normally associated with the Lie groups. This mathematical object was previously called an infinitesimal group.
    • Lattices: they are algebraic structures used in algebra and order theory; their name comes from the shape of Hasse diagrams.
    • Boolean algebras: algebraic structures that outline logical operations, used both in mathematics and in digital electronics and computer science.

What is the difference between abstract algebra and elementary algebra?

While elementary and abstract algebra both respond to the same general approaches to algebra, there are differences between them; for example, while elementary algebra is based on solving simple algebraic equations, abstract algebra looks at algebraic systems and structures or groups with different operations. Elementary algebra studies real numbers and complex numbers, while abstract algebra expresses mathematical structures that cannot necessarily be expressed with numerical values.

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