Identity Elements: The Neutral Value in Operations

In the context of a set (or Type) A and a binary operation • defined on that set (as discussed for semigroups), an identity element (often called a “unit element”) is a special value within A that, when combined with any other value from A using •, leaves that other value unchanged.

Defining the Identity Element

A value e (which must be an element of set A, so e has type 'a if A corresponds to type 'a) is called an identity element for the operation • on set A if, for every element x in A:

• Left identity property: e • x = x
• Right identity property: x • e = x

This must hold true for all possible values of x within the set A.

Identity Elements in Familiar Operations

Let’s look at some familiar binary operations and their identity elements:

Addition and Multiplication on Integers

For the Type int (the set of integers):

• With the binary operation (+) (addition):

  • The identity element is 0 (which is of type int).
  • Property: For any integer x, 0 + x = x and x + 0 = x.
    • Example: 0 + 3 = 3 and 3 + 0 = 3.

• With the binary operation (*) (multiplication):

  • The identity element is 1 (which is of type int).
  • Property: For any integer x, 1 * x = x and x * 1 = x.
    • Example: 1 * 5 = 5 and 5 * 1 = 5.

String Concatenation

For the Type string (the set of all strings):

• With the binary operation (+) (string concatenation):
  • The identity element is "" (the empty string, which is of type string).
  • Property: For any string s, "" + s = s and s + "" = s.
    • Example: "" + "hello" = "hello" and "hello" + "" = "hello".

Physical Examples: The Challenge of Finding a Natural Identity

In our physical analogies, a true identity element in the mathematical sense is often hard to find or doesn’t naturally exist:

• LEGO blocks (Set: LEGO blocks, Operation: joining two blocks):

  • Is there a LEGO block e such that joining e with any block X results in X unchanged (i.e., e joined with X is indistinguishable from X alone, and X joined with e is also indistinguishable from X alone)?
  • No, any physical LEGO piece, when connected, physically alters the resulting structure or occupies space. There isn’t a “do-nothing” LEGO block that leaves the other block as if no connection was made.

• USB devices (Set: USB devices, Operation: connecting two devices):

  • Similarly, there’s no physical USB device e that, when connected to another device X (or a hub X), results in a configuration that is functionally and structurally identical to X alone. Any connection typically introduces at least a new point in the device tree.

Uniqueness and Two-Sidedness of Identity Elements

An important property of an identity element, if one exists for a given binary operation • on a set A, is that it must be unique and it must be both a left identity and a right identity.

Let’s prove this:

Assume, for a set A with a binary operation •, that eL is a left identity (so eL • x = x for all x in A) and eR is a right identity (so x • eR = x for all x in A). Both eL and eR are elements of A.

1. Since eL is a left identity, it holds for x = eR. So, eL • eR = eR.
2. Since eR is a right identity, it holds for x = eL. So, eL • eR = eL.

From steps 1 and 2, we have eR = eL • eR and eL = eL • eR. Therefore, eL = eR.

This shows that if a left identity and a right identity both exist, they must be the same element. This also implies that if an identity element exists, it is unique. (If e1 and e2 were both two-sided identities, then e1 = e1 • e2 (since e2 is a right identity) and e1 • e2 = e2 (since e1 is a left identity), thus e1 = e2.)

So, for a given binary operation on a set:

• You can’t have just a left identity that isn’t also a right identity (if a right identity also exists, and vice-versa).
• If an identity element exists, it’s the only one.

Let’s confirm with our examples:

• For (int, +): 0 is the unique two-sided identity.
• For (int, *): 1 is the unique two-sided identity.
• For (string, +): "" is the unique two-sided identity.

This uniqueness and two-sided nature is a fundamental characteristic of identity elements in algebraic structures.