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Identity element

In abstract algebra, an identity element (also called a neutral element) for a binary operation on a set is an element e that, when combined with any other element a in the set using the operation, yields a itself.^[1] Formally, e satisfies a * e = e * a = a for all a in the set, where * denotes the binary operation.^[2] If an identity element exists for a binary operation on a nonempty set, it is unique.^[3] To see this, suppose e and e' both serve as identities; then e * e' = e' (since e is an identity) and e * e' = e (since e' is an identity), implying e = e'.^[3] Not every binary operation has an identity—for instance, the operation x * y = 1 + xy on the integers lacks one.^[2] Common examples include 0 as the additive identity for the integers under addition, since n + 0 = 0 + n = n for any integer n, and 1 as the multiplicative identity under multiplication, since n \times 1 = 1 \times n = n.^[1] Another case is the operation x * y = x + y + 1 on the integers, where -1 acts as the identity.^[2] The identity element plays a foundational role in algebraic structures such as monoids and groups, where its existence is a defining axiom that enables concepts like inverses and ensures operational consistency.^[3]

Definitions

Basic Definition

In mathematics, particularly in abstract algebra, an identity element is defined within the context of a set equipped with a binary operation. Consider a non-empty set S and a binary operation * on S, which is a function from the Cartesian product S \times S to S, meaning that for every pair of elements a, b \in S, the result a * b is also in S (a property known as closure). This setup forms a binary structure \langle S, * \rangle, where the operation is well-defined but no further properties, such as associativity, are assumed at this stage.^[4] An element e \in S is called an identity element (or neutral element) for the binary structure \langle S, * \rangle if it satisfies the two-sided condition: for all a \in S,

a * e = e * a = a.

This formal requirement ensures that e interacts with every element in a way that preserves the original element under the operation from either side, establishing the standard definition of a two-sided identity.^[4] The identity element serves as a neutral counterpart to the operation, leaving other elements unchanged when combined with them, which facilitates the analysis of more complex algebraic behaviors in structures that possess such an element. While not every binary structure contains an identity, its presence defines a unital magma.^[4]

Left and Right Identities

In algebraic structures such as magmas, a left identity is an element e in the set S with respect to a binary operation * such that e * a = a for all a \in S, with no requirement that a * e = a.^[5] Similarly, a right identity is an element e \in S such that a * e = a for all a \in S, without any condition on e * a.^[5] An element that serves as both a left identity and a right identity is termed a two-sided identity, satisfying e * a = a * e = a for all a \in S; however, the existence of a one-sided identity does not guarantee the other side, allowing for cases where left or right identities appear independently in non-standard operations.^[6] For instance, consider the infinite set of natural numbers with the operation x * y = y, which is associative and renders every element a left identity (x * y = y) but admits no right identity.^[5]

Properties

Uniqueness

In algebraic structures equipped with a binary operation, the identity element, if it exists, is unique. To see this, suppose e and f are both identity elements for the operation \cdot on a set S, meaning a \cdot e = e \cdot a = a and a \cdot f = f \cdot a = a for all a \in S. Then, e = e \cdot f = f, where the first equality uses the right-identity property of f and the second uses the left-identity property of e.^[7] This proof follows directly from the definition of a two-sided identity and requires no additional axioms beyond the existence of the operation itself. It applies to any magma (a set with a binary operation) that possesses such an element. For one-sided identities, uniqueness holds under certain conditions. Specifically, if a magma has both a left identity l (satisfying l \cdot m = m for all m \in S) and a right identity r (satisfying m \cdot r = m for all m \in S), then l = l \cdot r = r, making it a unique two-sided identity. However, a left identity alone may not be unique.^[7] The uniqueness of the identity element has significant implications for classifying algebraic structures. For example, in the definition of a monoid—a magma with an associative operation and an identity—the presence of the identity guarantees exactly one such element, facilitating the study of further properties like inverses and homomorphisms.^[8]

Behavior in Algebraic Structures

In a monoid (M, \cdot), the identity element e is required by definition as the unique neutral element that satisfies e \cdot a = a \cdot e = a for all a \in M, with the binary operation \cdot being associative.^[9] This structure ensures that e acts as a fixed point under the operation, enabling consistent composition without altering elements.^[10] The associativity of \cdot guarantees that e interacts uniformly in any parenthesized expression, preserving the monoid's operational integrity.^[11] Groups extend monoids by incorporating inverses relative to the identity: for every a \in G, there exists a^{-1} \in G such that a \cdot a^{-1} = a^{-1} \cdot a = e, where (G, \cdot) is associative with e as the neutral element.^[12] This inverse property ties e directly to solvability, as equations like a \cdot x = e have unique solutions x = a^{-1}, facilitating cancellation and the resolution of operational equations within the group.^[13] In rings (R, +, \cdot), distinct identities exist for each operation: the additive identity $0satisfiesa + 0 = 0 + a = afor alla \in R

, forming an abelian group under addition, while a multiplicative identity &#36;1

(in unital rings) satisfies a \cdot 1 = 1 \cdot a = a.^[14] Distributivity (a \cdot (b + c) = a \cdot b + a \cdot c and (b + c) \cdot a = b \cdot a + c \cdot a) links the operations but leaves the identities' neutral roles unchanged.^[15] The identity element's presence enables homomorphisms between structures: a monoid homomorphism f: (M, \cdot) \to (N, \star) preserves the operation (f(a \cdot b) = f(a) \star f(b)) and maps e_M to e_N, similarly for group and ring homomorphisms which also preserve inverses or distributivity as applicable.^[16] This preservation allows identities to serve as anchors for mapping structural properties, such as isomorphisms or quotients.^[17] In non-unital structures like semigroups, no identity is required, consisting solely of an associative operation without a neutral element, contrasting with monoids and highlighting the identity's optional yet foundational role in more complete algebraic frameworks.^[13]

Examples

In Numerical Operations

In the context of numerical operations on the real numbers, the additive identity element is 0, meaning that for any real number a, a + 0 = 0 + a = a.^[18] This property holds for integers as well, where addition forms an abelian group with 0 as the unique identity.^[19] For multiplication on the real numbers, the identity element is 1, such that a \times 1 = 1 \times a = a for any real number a.^[20] However, in the multiplicative structure of the real numbers excluding zero, which forms a group, every non-zero element has an inverse, but zero itself lacks a multiplicative inverse in the full set of reals.^[21] Subtraction on the real numbers does not possess a global two-sided identity element, as there is no number e satisfying a - [e](/page/e) = [e](/page/e) - a = a for all a; while 0 acts as a right identity since a - [0](/page/0) = a, it fails as a left identity because $0 - a = -a \neq a unless a = [0](/page/0)./05%3A_Sample_Topics/5.02%3A_Abstract_Algebra-_commutative_groups) Similarly, division lacks a global identity due to domain restrictions (division by zero is undefined) and the absence of a two-sided neutral element; 1 serves only as a right identity, as a / [1](/page/1) = a but $1 / a \neq a for a \neq [1](/page/1).^[22] In vector spaces over the real numbers, the zero vector \vec{0} functions as the additive identity, ensuring \vec{v} + \vec{0} = \vec{0} + \vec{v} = \vec{v} for any vector \vec{v}.^[18] This aligns with the broader role of the zero element in additive groups underlying such structures.^[23]

In Set Theory

In set theory, the identity element for binary operations on sets, particularly within the power set of a given universe, behaves as a neutral element that leaves any operand unchanged. For the union operation, denoted ∪, the empty set ∅ serves as the identity element. Specifically, for any set A, A ∪ ∅ = ∅ ∪ A = A, meaning that adjoining the empty set to any set via union yields the original set unchanged.^[24] For the intersection operation, denoted ∩, the identity element is the universal set U, which encompasses all elements under consideration in the given context. Thus, for any set A ⊆ U, A ∩ U = U ∩ A = A, preserving the set A when intersected with the full universe. This identity is relative to the chosen universe U, as intersections are typically defined within a bounded domain.^[24] The symmetric difference operation, denoted Δ and defined as A Δ B = (A ∪ B) \ (A ∩ B), also has the empty set ∅ as its identity element. For any set A, A Δ ∅ = ∅ Δ A = A, since the symmetric difference with the empty set excludes no elements from A and adds none. This operation forms a monoid on the power set with ∅ as the neutral element.^[24] In contrast, the Cartesian product ×, which produces ordered pairs from elements of two sets, lacks a universal two-sided identity element across the class of all sets. While the empty set ∅ satisfies A × ∅ = ∅ for any A (acting as a sort of absorbing element rather than identity), no set E exists such that A × E = E × A = A for all A, as assuming such an E leads to contradictions (e.g., ∅ × E = ∅ implies E = ∅, but ∅ × {x} ≠ {x}). Thus, the Cartesian product does not admit a neutral element in the standard sense.^[25]

In Functions and Transformations

In the context of function composition, the identity element is the identity function, denoted id, defined by id(x) = x for all x in its domain. This function acts as the neutral element in the monoid of functions under composition, satisfying f \circ id = id \circ f = f for any function f from a set to itself.^[26]^[27] For matrix multiplication, the identity element is the identity matrix I, an n \times n square matrix with 1s along the main diagonal and 0s elsewhere. It satisfies A I = I A = A for any n \times n matrix A, serving as the multiplicative identity in the ring of matrices. For example, the 2×2 identity matrix is

\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.

This property ensures that multiplying by I leaves the matrix unchanged, analogous to the number 1 in scalar multiplication.^[28] In groups of symmetries, such as the dihedral group representing rotations and reflections of a regular polygon, the identity transformation—often described as the "do-nothing" operation—leaves every point in the space fixed and functions as the neutral element under composition. Composing any symmetry transformation g with the identity e yields g \circ e = e \circ g = g, preserving the group's structure.^[29]^[30] The concept extends to permutations, where the symmetric group S_n comprises all bijections on a set of n elements, with composition as the operation. The identity permutation, which maps each element to itself (e.g., in cycle notation as (1)(2)\cdots(n)), is the neutral element, satisfying \sigma \circ id = id \circ \sigma = \sigma for any permutation \sigma \in S_n. This makes S_n a group of order n!.^[31]

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