📂Abstract Algebra

Binary Operations in Abstract Algebra

Buildup

Mathematics can be broadly divided into three categories: geometry, analysis, and algebra. Among these, algebra was a branch of mathematics dealing with binomials, reduction, etc., in the curriculum. Algebra essentially aimed to solve any equation using letters instead of numbers. It sought after a general and powerful method of solution, not limited to specific numbers, thus could be considered cutting-edge technology of the time. However, these mathematical techniques have become common knowledge to everyone in the modern era of developed education.

Meanwhile, the mathematical community started to develop these concepts further, moving beyond ’numbers’ to focus on abstract ‘structures’. What we previously called ’numbers’ and ‘calculations’ has been abstracted to ’elements’ and ‘operations’. Therefore, modern algebra has become a discipline that studies the conditions under which algebraic techniques can be used or the structures themselves. As can be guessed from the above explanation, modern algebra is especially abstract, often referred to as ‘abstract algebra’.

Abstract algebra primarily focuses on the properties of certain sets and the operations defined on those sets. Given a set $S$ and an operation $\ast$, it studies whether $S$ is closed under $\ast$, whether there is an identity element, whether there are inverses, etc. Among these, the operations of interest in abstract algebra are binary operations, where two elements correspond to one element like $a \ast\ b = c$.

Definition ¹

1. Binary operation can be considered as a function defined as $\ast : S \times S \to S$, and the set on which such a binary operation is defined is called a binary operation structure.
2. For an element $a,b$ in a set $M \ne \emptyset$ and a binary operation $\ast$, if $a * b \in M$ then $\left< M , \ast\ \right>$ is defined as a magma.

Description

A magma is the simplest concept among the binary operation structures of interest in abstract algebra. It just needs to be closed.

Examples of not being a magma

The set of odd numbers $O$ and the set of irrational numbers $I$ are not magmas.

Examples that cannot be magmas include the set of odd numbers and the set of irrational numbers. These are not closed under operations like multiplication or addition, and so they cannot be magmas despite being binary operation structures.

• Considering the addition for the set of odd numbers $O$, the sum of two odd numbers is always even, so $O$ is not closed and cannot be a magma.
• Considering multiplication for the set of irrational numbers $I$, $\sqrt{2} , 2\sqrt{2} \in I$ and $\sqrt{2} \cdot 2 \sqrt{2 } = 4 $, but since $ 4 \notin I$, $I$ is not a magma.

Examples of being a magma

The power set $\mathscr{P}(S)$ of any set $S$ and the difference $\setminus$ are magmas.

• For subsets $A$ and $B$ of $S$, $( A \setminus B ) \subset S$, thus $( A \setminus B ) \in \mathscr{P}(S)$, and $\left< \mathscr{P}(S) , \setminus \right>$ is a magma.

Operations Matter

One important thing is that when exploring algebraic structures, not only the set itself but also the operations need to be considered. Let’s revisit the examples that are not magmas.

The set of odd numbers $O$, and the set of irrational numbers $I$, $\left< O , \cdot \right>$ is a magma but $\left< I , + \right>$ is not.

• Considering multiplication for the set of odd numbers $O$, the product of two odd numbers is always odd, so $O$ is closed and is a magma.
• Considering addition for the set of irrational numbers $I$, $\sqrt{2} , -\sqrt{2} \in I$ and $\sqrt{2} + ( - \sqrt{2 } ) = 0$, but since $ 0 \notin I$, $I$ is not a magma.

Although the set of odd numbers became a magma with a different operation, the set of irrational numbers still could not be a magma. In essence, even sets that seem meaningless at the moment can have the potential to hold meaningful algebraic structures, depending on the operation given.

On the other hand, since the definition of magma is so simple and general, magma itself does not provide useful properties. The term magma, sharing the same root as the ’lava’ we know, means ‘mixed bag’ in French. It signifies that many algebraic structures start as magmas, but the concept itself is not critically important.