Algebraic structure facts for kids

An algebraic structure is a special way to organize a collection of items, like numbers or symbols. Imagine you have a group of things (mathematicians call this a set). An algebraic structure happens when you can do something with these items, like adding or multiplying them. This "doing something" is called an operation.

Think of it like a game with rules. The set is your collection of game pieces, and the operation is how you move or combine those pieces. Different rules make different kinds of algebraic structures.

What is a Binary Operation?

A binary operation is a rule that takes two items from your set and combines them to make a third item, which is also part of the same set.

For example:

• Addition (+) is a binary operation on whole numbers. If you take 3 and 5, you get 8, which is also a whole number.
• Multiplication (x) is also a binary operation. If you take 2 and 4, you get 8.

Structures with One Operation

Here are some basic algebraic structures that use just one binary operation:

• Magma

* A magma is the simplest structure. It's just a set (your collection of items) and one binary operation (a rule for combining two items). There are no other special rules for how the operation works.

• Semigroup

* A semigroup is a magma where the operation is associative. This means that if you combine three items, it doesn't matter how you group them. * For example, with addition: (2 + 3) + 4 is the same as 2 + (3 + 4). Both equal 9.

• Monoid

* A monoid is a semigroup that has an special item called an identity element. This identity element doesn't change other items when combined with them using the operation. * For addition, the identity element is 0, because 5 + 0 = 5. * For multiplication, the identity element is 1, because 5 x 1 = 5.

• Group

* A group is a monoid where every item has an inverse element. An inverse element is another item that, when combined with the first item using the operation, gives you the identity element. * For addition, the inverse of 5 is -5, because 5 + (-5) = 0 (the identity element). * For multiplication, the inverse of 5 is 1/5, because 5 x (1/5) = 1 (the identity element).

• Commutative Group

* A commutative group (also called an Abelian group) is a group where the operation is commutative. This means the order of the items doesn't matter when you combine them. * For example, with addition: 2 + 3 is the same as 3 + 2. Both equal 5. * However, not all operations are commutative. For example, subtraction (5 - 2 is not the same as 2 - 5).

Structures with Two Operations

Some algebraic structures use two different binary operations, usually called "addition" and "multiplication." These operations have to work together in specific ways.

• Ring

* A ring is a set with two operations: addition and multiplication. * With addition, the set forms a commutative group. * With multiplication, the set forms a semigroup (or sometimes a monoid, depending on the exact definition). * The two operations must also follow the distributive property. This means that multiplying a number by a sum is the same as multiplying it by each part of the sum and then adding the results. * For example: 2 x (3 + 4) = (2 x 3) + (2 x 4). Both sides equal 14.

• Commutative Ring

* A commutative ring is a ring where the multiplication operation is also commutative (the order doesn't matter). * For example, with regular numbers, 2 x 3 is the same as 3 x 2.

• Field

* A field is a commutative ring where the set, with multiplication (excluding the additive identity, usually 0), also forms a group. This means that every item (except zero) has a multiplicative inverse. * Fields are very important in mathematics because they behave a lot like the numbers we use every day.

Examples of Algebraic Structures

Let's look at some common number sets and see what kind of algebraic structures they form:

• The whole numbers (0, 1, 2, 3, ...) with addition form a monoid. They have an identity element (0), and addition is associative. But they don't form a group because negative numbers (like -5) are not whole numbers, so 5 doesn't have an additive inverse within the set.
• The integers (..., -2, -1, 0, 1, 2, ...) with addition form a commutative group. They have 0 as an identity, every integer has a negative inverse (e.g., the inverse of 5 is -5), and addition is associative and commutative.
• The integers with multiplication form only a monoid. They have 1 as an identity, and multiplication is associative. But most integers don't have a multiplicative inverse that is also an integer (e.g., the inverse of 5 is 1/5, which is not an integer).
• The integers with both addition and multiplication form a commutative ring. They satisfy all the rules for a commutative ring, but they are not a field because, as mentioned, most integers don't have multiplicative inverses within the set.
• The rational numbers (fractions), the real numbers (all numbers on the number line), and the complex numbers (numbers involving the square root of -1) with their usual addition and multiplication are all examples of fields. This is because every number (except zero) in these sets has a multiplicative inverse.

See also

In Spanish: Estructura algebraica para niños