Commutativity and Associativity in Binary Operations: Exploring the Independence of These Properties

Understanding Commutativity and Associativity

Commutativity and associativity are two fundamental properties of binary operations that play crucial roles in the structure of mathematical systems. These properties, while distinct, are often interrelated in various algebraic structures. However, one does not necessarily imply the other. This article delves into the distinction and independence of these properties through various definitions, examples, and explanations.

Definitions

The commutativity of a binary operation is a property that states the order of the operands does not affect the outcome. Formally, a binary operation ° on a set is commutative if for all elements (mathbf{a}) and (mathbf{b}) in that set, the equation (mathbf{a} ° mathbf{b} mathbf{b} ° mathbf{a}) holds. On the other hand, associativity refers to the property where the grouping of operands does not change the result. A binary operation ° on a set is associative if for all elements (mathbf{a}), (mathbf{b}), and (mathbf{c}) in the set, the equation ((mathbf{a} ° mathbf{b}) ° mathbf{c} mathbf{a} ° (mathbf{b} ° mathbf{c})) holds.

Examples of Commutative and Associative Operations

Commutative and Associative Operations

Consider the binary operation of addition and multiplication on the real numbers:

For addition: (mathbf{a} mathbf{b} mathbf{b} mathbf{a}) and ((mathbf{a} mathbf{b}) mathbf{c} mathbf{a} (mathbf{b} mathbf{c})). For multiplication: (mathbf{a} times mathbf{b} mathbf{b} times mathbf{a}) and ((mathbf{a} times mathbf{b}) times mathbf{c} mathbf{a} times (mathbf{b} times mathbf{c})).

A Non-Commutative and Non-Associative Operation

Consider the operation defined by (mathbf{a} circ mathbf{b} mathbf{a} - mathbf{b}):

This operation is not commutative: (mathbf{a} - mathbf{b} eq mathbf{b} - mathbf{a}) in general. It is also not associative: ((mathbf{a} - mathbf{b}) - mathbf{c} eq mathbf{a} - (mathbf{b} - mathbf{c})) in general.

Conclusion: Commutativity Does Not Imply Associativity

From the above examples, we can see that a binary operation can be commutative without being associative or vice versa. This means that neither property necessarily implies the other.

General Principles and Further Considerations

The operations in groups, such as the set of integers with addition, are all associative, implying that commutativity in groups means that associativity is inherently satisfied. However, for a magma (a set with a binary operation), the situation is different. A magma's multiplication may be commutative without being associative.

A Non-Associative Example: The Average Operation

Consider the average operation on rational numbers defined as (mathbf{x} circ mathbf{y} frac{mathbf{x} mathbf{y}}{2}):

Commutativity is apparent: (mathbf{x} circ mathbf{y} mathbf{y} circ mathbf{x}), because (mathbf{x} mathbf{y} mathbf{y} mathbf{x}). However, ((mathbf{0} circ mathbf{0}) circ mathbf{4} mathbf{0} circ mathbf{2} mathbf{1}), but (mathbf{0} circ (mathbf{0} circ mathbf{4}) mathbf{0} circ mathbf{2} mathbf{2}), which shows that associativity fails.

Intuitively, taking the average of three things should be adding them up and dividing by three, but in this example, the order of averaging matters. This demonstrates that commutativity alone does not guarantee associativity.

There are many other examples of commutative non-associative magmas. In particular, there is a class of algebras called Jordan algebras, which are both commutative and non-associative. These algebras provide a rich structure that further emphasizes the independence of commutativity and associativity.

Conclusion

In summary, commutativity and associativity are distinct properties that do not necessarily imply one another. While they are often found together in certain algebraic structures, such as groups, they can be mutually exclusive in other settings, like magmas. This independence is a fundamental concept in abstract algebra and has profound implications for the structure and classification of algebraic systems.