Algebraic structure

 

Basic concepts[edit]

Main article: Algebraic structure

By abstracting away various amounts of detail, mathematicians have defined various algebraic structures that are used in many areas of mathematics. For instance, almost all systems studied are sets, to which the theorems of set theory apply. Those sets that have a certain binary operation defined on them form magmas, to which the concepts concerning magmas, as well those concerning sets, apply. We can add additional constraints on the algebraic structure, such as associativity (to form semigroups); identity, and inverses (to form groups); and other more complex structures. With additional structure, more theorems could be proved, but the generality is reduced. The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the corresponding theories: for instance, the theorems of group theory may be used when studying rings (algebraic objects that have two binary operations with certain axioms) since a ring is a group over one of its operations. In general there is a balance between the amount of generality and the richness of the theory: more general structures have usually fewer nontrivial theorems and fewer applications.

Algebraic structures between magmas and groups. For example, monoids are semigroups with identity.

Examples of algebraic structures with a single binary operation are:

• Magma
• Quasigroup
• Monoid
• Semigroup
• Group

Examples involving several operations include:

• Ring
• Field
• Module
• Vector space
• Algebra over a field
• Associative algebra
• Lie algebra
• Lattice
• Boolean algebra

Branches of abstract algebra[edit]

Group theory[edit]

Main article: Group theory

A group is a set ${\displaystyle �}$ together with a "group product", a binary operation ${\displaystyle \cdot :�\times �\to �}$. The group satisfies the following defining axioms:

Identity: there exists an element ${\displaystyle �}$ such that, for each element ${\displaystyle �}$ in ${\displaystyle �}$, it holds that ${\displaystyle �\cdot �=�\cdot �=�}$.

Inverse: for each element ${\displaystyle �}$ of ${\displaystyle �}$, there exists an element ${\displaystyle �}$ so that ${\displaystyle �\cdot �=�\cdot �=�}$.

Associativity: for each triplet of elements ${\displaystyle �,�,�}$ in ${\displaystyle �}$, it holds that ${\displaystyle (�\cdot �)\cdot �=�\cdot (�\cdot �)}$.

Ring theory[edit]

Main article: Ring Theory

A ring is a set ${\displaystyle �}$ together with two binary operations, addition: ${\displaystyle +:�\times �\to �}$ and multiplication: ${\displaystyle \cdot :�\times �\to �}$. Additionally, ${\displaystyle �}$ satisfies the following defining axioms:

Addition: ${\displaystyle �}$ is a commutative group under addition.

Multiplication: ${\displaystyle �}$ is a monoid under multiplication.

Distributive: Multiplication is distributive with respect to addition.

Applications[edit]

Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance, algebraic topology uses algebraic objects to study topologies. The Poincaré conjecture, proved in 2003, asserts that the fundamental group of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not. Algebraic number theory studies various number rings that generalize the set of integers. Using tools of algebraic number theory, Andrew Wiles proved Fermat's Last Theorem.

In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations. In gauge theory, the requirement of local symmetry can be used to deduce the equations describing a system. The groups that describe those symmetries are Lie groups, and the study of Lie groups and Lie algebras reveals much about the physical system; for instance, the number of force carriers in a theory is equal to the dimension of the Lie algebra, and these bosons interact with the force they mediate if the Lie algebra is nonabelian.^[50]