Algebraic Structure

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Definition

An Algebraic Structure is a mathematical concept that consists of a set together with a binary operation and operations on that set that satisfy certain axioms, known as the Laws of the structure. These Laws define how elements of the set interact with each other using the binary operation.

Types of Algebraic Structures

There are several types of algebraic structures, including:

• Group: A Group is an Algebraic Structure that consists of a set and a binary operation that satisfies four properties: Closure, Associativity, Identity Element, and Inverse Elements.
• Ring: A Ring is a generalization of the Group, where the additive identity is replaced by an additive inverse and multiplication is allowed to be commutative.
• Field: A Field is a special type of Ring in which every non-zero element has a multiplicative inverse.
• Vector Space: A Vector Space is an Algebraic Structure that consists of a set of vectors with operations for addition and scalar multiplication.

Laws of Algebraic Structures

The Laws of an Algebraic Structure define how the elements interact with each other using the binary operation. These Laws are:

Group Laws

1. Closure: For all elements x, y in the set, the result of the binary operation (x * y) is also in the set.
2. Associativity: For all elements x, y, z in the set, ((x * y) * z) = (x * (y * z)).
3. Identity Element: There exists an element e in the set such that for all elements x in the set, x * e = e * x = x.
4. Inverse Elements: For each element x in the set, there exists an element y in the set such that x * y = y * x = e.

Ring Laws

1. Commutativity of Addition: For all elements x and y in the set, x + y = y + x.
2. Associativity of Addition: For all elements x, y, z in the set, (x + y) + z = x + (y + z).
3. Distributivity: For all elements x, y, z in the set, x * (y + z) = x * y + x * z.
4. Existence of Multiplicative Identity: There exists an element 1 in the set such that for all elements x in the set, x * 1 = 1 * x = x.

Field Laws

1. Commutativity of Addition and Multiplication: For all elements x and y in the set, x + y = y + x and x * y = y * x.
2. Associativity of Addition and Multiplication: For all elements x, y, z in the set, (x + y) + z = x + (y + z) and (x * y) * z = x * (y * z).
3. Distributivity: For all elements x, y, z in the set, x * (y + z) = x * y + x * z.
4. Existence of Multiplicative Identity: There exists an element 1 in the set such that for all elements x in the set, x * 1 = 1 * x = x.

Examples

• The set of integers with addition and multiplication as binary operations satisfies the Group Laws.
• The set of real numbers with addition and multiplication as binary operations satisfies the Ring Laws.
• The set of non-zero vectors with scalar multiplication as an operation that satisfies commutativity, Associativity, and Distributivity is a Vector Space.

References

• “Algebra” by Serge Lang
• “Abstract Algebra” by David S. Dummit and Richard M. Foote