Binary Operations

Introduction

Let A be a set and let n ≥ 0 be an integer. An operation ♢ in A with n operands is a mapping

♢ : An → A

An operation with n operands is also called an operation with the arity n or an n-ary operation. For n = 0, 1, 2, 3, 4 one uses the special terms 0-ary, unary, binary, ternary and quaternary.

Constant operations

An operation ♢ in A with 0 operands is uniquely determined by ♢ (()) that is a certain element of A, which the operation can be identified with. The operations with 0 operands are therefore sometimes called the constant operations. So we shall all the time think of an operation with 0 operands as an element of A.

Operation symbols

To underline the general aspect we are going to use operation symbols which do not already have a definite meaning such as x ♢ y, x ♡ y. There are a number of alternative ways to denote an operation, other than

♢ (a1, . . ., an)

namely

♢ a1 . . . an, a1 ♢ . . . ♢ an, a1 . . . an

called respectively prefix, infix and postfix notation. In some contexts we do not state which notation we use in the hope that it is otherwise easily understood from the context. For unary operations only prefix and postfix notation are relevant. Generally we use the operator notation. Infix notation is mostly used for binary operations.

Associative

A binary operation ♢ on A is associative if

a ♢ (b ♢ c) = (a ♢ b) ♢ c

for all a, b, c ∈ A

Commutative

A binary operation ♢ on A is commutative if

a ♢ b = b ♢ a

for all a, b ∈ A

Distributive

A binary operation ♢ is distributive w.r.t. a binary operation ♡ if

a ♢ (b1 ♡ b2) = (a ♢ b1) ♡ (a ♢ b2)

for all a, b ∈ A

An algebraic structure is a set equipped with operations in the set. If the set itself is A and if the operations are the ordered tuple ( ♢1 , . . ., ♢n) we shall use

(A, ♢1, . . ., ♢n)

to denote the structure.