Abstract Algebra - Basic Concepts

Basic Concepts

Abstract algebra studies the properties and patterns that seemingly disparate mathematical concepts have in common. For example, consider the distinct operations of function composition, f(g(x)), and of matrix multiplication, AB. These two operations have, in fact, the same structure. To see this, think about multiplying two square matrices, AB, by a one column vector, x. This defines a function equivalent to composing Ay with Bx: Ay = A(Bx) = (AB)x. Functions under composition and matrices under multiplication are examples of monoids. A set S and a binary operation over S, denoted by concatenation, form a monoid if the operation associates, (ab)c = a(bc), and if there exists an e ∈ S, such that ae = ea = a.

Another example of two different systems having similar algebraic structure is the set of 90 degree rotations and the set {1, i, -1, -i} under multiplication. Notice that rotating an object by 90 degrees twice is the same as rotating by 180 degrees; similarly, i*i=-1. In fact, by replacing 0-degree rotations by 1, 90-degree rotations by i, 180-degree rotations by -1, and 270-degree rotations by -i, the set of rotations is transformed into the set of {1, i, -1, -i} under multiplication; these two objects have the same algebraic structure called a group.

By abstracting away various amounts of detail, mathematicians have created theories of various algebraic structures that apply to many objects. 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); associativity, 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 apply to rings (algebraic objects that have two binary operations with certain axioms) since a ring is a group over one of its operations. Mathematicians choose a balance between the amount of generality and the richness of the theory.

Examples of algebraic structures with a single binary operation are:

• Magmas
• Quasigroups
• Monoids
• Semigroups
• Groups

More complicated examples include:

• Rings
• Fields
• Modules
• Vector spaces
• Algebras over fields
• Associative algebras
• Lie algebras
• Lattices
• Boolean algebras