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Homomorphism

In mathematics, particularly within abstract algebra, a homomorphism is a function between two algebraic structures of the same type—such as groups, rings, or vector spaces—that preserves the operations and relations defining those structures.^[1] For instance, in the case of groups G and H, a homomorphism \phi: G \to H satisfies \phi(g_1 g_2) = \phi(g_1) \phi(g_2) for all g_1, g_2 \in G, ensuring compatibility with the group multiplication.^[2] Similarly, for rings R and S, it preserves both addition and multiplication: \phi(a + b) = \phi(a) + \phi(b) and \phi(ab) = \phi(a) \phi(b) for all a, b \in R.^[3] Key properties of homomorphisms include their images and kernels, which provide insights into the structural relationships between the domain and codomain. The image of a homomorphism \phi: G \to H is the subset \phi(G) \subseteq H, which forms a subgroup of H.^[2] The kernel is the preimage of the identity element in H, defined as \ker(\phi) = \{ g \in G \mid \phi(g) = e_H \}, and it is always a normal subgroup of G.^[2] These elements enable the formation of quotient structures; for example, the first isomorphism theorem states that G / \ker(\phi) \cong \operatorname{im}(\phi), linking the original group to a simpler isomorphic copy.^[4] Homomorphisms are foundational to understanding algebraic similarities and classifications, as they reveal how one structure can be embedded into or projected onto another. A bijective homomorphism, called an isomorphism, establishes that two structures are essentially identical up to relabeling, while injective ones (monomorphisms) allow embeddings as substructures.^[1] Examples abound in applications: the exponential map x \mapsto e^x is a homomorphism from (\mathbb{R}, +) to (\mathbb{R}^+, \times), and the determinant function is a homomorphism from the general linear group GL_n(\mathbb{R}) to the multiplicative group \mathbb{R}^\times.^[2] Beyond groups and rings, homomorphisms extend to modules, fields, and categories, underpinning theorems in Galois theory and representation theory.^[1]

Core Concepts

Definition

In category theory, a homomorphism is synonymous with a morphism, which is an arrow f: A \to B between objects A and B in a category. Categories consist of objects (such as sets or algebraic structures) and morphisms (maps between them) with associative composition and identity morphisms obeying the category axioms.^[5]^[6] In algebra, a homomorphism is a function f: S \to T between two algebraic structures S and T of the same type that preserves their operations; for instance, in groups (S, \cdot_S) and (T, \cdot_T), it satisfies f(a \cdot_S b) = f(a) \cdot_T f(b) for all a, b \in S, and similarly for rings (preserving addition and multiplication) or vector spaces (preserving addition and scalar multiplication).^[7]^[5] Note that for rings, some definitions require homomorphisms to preserve the multiplicative identity (unital ring homomorphisms), while others do not. For algebraic structures with compatible identity elements, such as groups, homomorphisms map the identity to the identity; similar conventions apply to unital rings and vector spaces. Standard notation employs f or \phi for these functions, with the preservation conditions ensuring the map respects the underlying algebraic relations without altering the intrinsic operations.^[5] These definitions presuppose familiarity with categories as collections of objects and morphisms, and algebraic structures like groups, rings, or vector spaces equipped with compatible operations.^[6] A bijective homomorphism whose inverse is also a homomorphism is termed an isomorphism, establishing structural equivalence between objects.^[8]

Properties

Homomorphisms possess fundamental preservation properties that maintain the structural integrity of the source and target objects. In the context of groups, a homomorphism \phi: G \to H maps the identity element of G to the identity element of H, i.e., \phi(e_G) = e_H.^[9] It also preserves inverses, satisfying \phi(g^{-1}) = \phi(g)^{-1} for all g \in G.^[10] These properties extend to other algebraic structures, where homomorphisms preserve the defining operations and relations; for instance, in partially ordered sets (posets), a homomorphism f: P \to Q is order-preserving, meaning if a \leq_P b then f(a) \leq_Q f(b).^[11] A key universal property of homomorphisms is their closure under composition. If \phi: G \to H and \psi: H \to K are homomorphisms between groups, then the composite map \psi \circ \phi: G \to K is also a homomorphism.^[2] This composition property underpins the formation of categories, where objects are the algebraic structures and morphisms are the homomorphisms.^[12] The first isomorphism theorem provides a structural relation between the kernel and image of a homomorphism (see Structural Elements). For a group homomorphism \phi: G \to H, there exists a natural isomorphism G / \ker(\phi) \cong \operatorname{im}(\phi), where \ker(\phi) is the preimage of the identity in H.^[13] In topological contexts, such as topological groups, homomorphisms are often continuous, thereby preserving limits and the topological structure.^[14]

Examples

Algebraic Examples

In group theory, a concrete example of a homomorphism is the projection map from the direct product group \mathbb{Z} \times \mathbb{Z} to \mathbb{Z}, defined by f(m, n) = m. This map preserves the group operation of addition: for any (m, n), (p, q) \in \mathbb{Z} \times \mathbb{Z}, f((m, n) + (p, q)) = f(m + p, n + q) = m + p = f(m, n) + f(p, q).^[15] In ring theory, the inclusion map from the ring of integers \mathbb{Z} to the field of rational numbers \mathbb{Q}, given by f(k) = k for all k \in \mathbb{Z}, is a ring homomorphism. It preserves both addition and multiplication: f(a + b) = a + b = f(a) + f(b) and f(a \cdot b) = a \cdot b = f(a) \cdot f(b), and it maps the multiplicative identity $1 \in \mathbb{Z} to $1 \in \mathbb{Q}.^[16] In the context of vector spaces over the real numbers, any linear transformation T: \mathbb{R}^n \to \mathbb{R}^m serves as a homomorphism, preserving vector addition and scalar multiplication: T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) and T(c \mathbf{u}) = c T(\mathbf{u}) for \mathbf{u}, \mathbf{v} \in \mathbb{R}^n and c \in \mathbb{R}. Such transformations admit a matrix representation: relative to standard bases, T(\mathbf{x}) = A \mathbf{x} where A is an m \times n matrix whose columns are the images of the standard basis vectors of \mathbb{R}^n.^[17]

Categorical Examples

In category theory, homomorphisms are precisely the morphisms of a category, which abstract the notion of structure-preserving maps across various mathematical domains. A foundational example occurs in the category Set, where the objects are sets and the morphisms—termed homomorphisms—are arbitrary functions between sets. Composition of these homomorphisms corresponds to the standard function composition, ensuring that the diagram of sets and functions satisfies the categorical axioms of associativity and identity preservation.^[18] Another illustrative case arises in the category of partially ordered sets, often denoted Poset, where objects are posets and homomorphisms are order-preserving maps. Such a map f: (P, \leq_P) \to (Q, \leq_Q) satisfies x \leq_P y implies f(x) \leq_Q f(y) for all x, y \in P, thereby maintaining the partial order structure under the mapping. These homomorphisms form the arrows of Poset, with composition defined pointwise as in Set, highlighting how categorical homomorphisms generalize relational preservation beyond algebraic operations.^[19] Homomorphisms can also be induced by functors, which are structure-preserving maps between categories themselves. A functor F: \mathcal{C} \to \mathcal{D} sends objects of \mathcal{C} to objects of \mathcal{D} and morphisms (homomorphisms) of \mathcal{C} to morphisms of \mathcal{D}, preserving composition and identities. For instance, the forgetful functor U: \mathbf{Grp} \to \mathbf{Set} from the category of groups to sets maps each group homomorphism \phi: G \to H to its underlying set function U(\phi): U(G) \to U(H), effectively "forgetting" the group operation while retaining the mapping. This induction underscores the role of functors in generating homomorphisms across categorical levels.^[20] Central to the structure of any category is the hom-set \operatorname{Hom}_{\mathcal{C}}(A, B), which collects all homomorphisms from an object A to an object B in \mathcal{C}. In locally small categories, these hom-sets form actual sets, and composition of homomorphisms defines a binary operation on them, endowing the category with its arrow framework. For example, in Set, \operatorname{Hom}_{\mathbf{Set}}(A, B) is the power set of all functions from A to B, illustrating how hom-sets encapsulate the relational essence of homomorphisms.^[21]

Special Types

Isomorphisms and Automorphisms

An isomorphism between two algebraic structures is a bijective homomorphism whose inverse function is also a homomorphism.^[22] This ensures that the structures are equivalent in a strong sense, preserving not only the operations but also allowing a reversible mapping between them.^[23] Such maps are often denoted by the symbol ≅, indicating that the structures are isomorphic.^[24] An automorphism is a special case of an isomorphism where the domain and codomain are the same structure, effectively a symmetry of the structure itself.^[25] The collection of all automorphisms of a structure forms a group under composition, known as the automorphism group.^[26] For instance, in the dihedral group D_n (n ≥ 3), which models the symmetries of a regular n-gon, automorphisms map the rotation subgroup \langle r \rangle to itself via f(r) = r^a where a is coprime to n, effectively "rotating" the generator of rotations while adjusting reflections accordingly.^[27] A concrete example of an isomorphism is the map \phi: \mathbb{Z} \to 2\mathbb{Z} defined by \phi(n) = 2n, which is a bijective group homomorphism under addition, with inverse \psi(m) = m/2 for even m, also a homomorphism.^[28] Another example arises in fields: the complex conjugation map \sigma: \mathbb{C} \to \mathbb{C} given by \sigma(a + bi) = a - bi (for a, b \in \mathbb{R}) is a field automorphism, as it is bijective and preserves addition and multiplication.^[29] Isomorphisms preserve all structural properties of the underlying algebraic structures, such as the orders of elements, the identity element, subgroups, and the overall group order.^[30] For groups specifically, if \phi: G \to H is an isomorphism, then \phi maps subgroups of G bijectively to subgroups of H, and the order of \phi(g) equals the order of g for each g ∈ G.^[31] This preservation underscores why isomorphic structures are considered indistinguishable algebraically.^[32]

Endomorphisms

An endomorphism of an algebraic structure S is a homomorphism \phi: S \to S that preserves the operations of S.^[33] This concept arises naturally in various algebraic categories, where it captures self-maps that respect the structure's axioms, such as group operations or ring multiplications.^[34] The set of all endomorphisms of S, denoted \operatorname{End}(S), forms a ring under pointwise addition and composition of maps as multiplication.^[35] Specifically, for an abelian group A, \operatorname{End}(A) consists of all group homomorphisms from A to itself, with addition defined by (\phi + \psi)(a) = \phi(a) + \psi(a) and multiplication by \phi \circ \psi.^[34] This ring structure endows endomorphisms with algebraic properties amenable to ring-theoretic analysis, such as units corresponding to automorphisms.^[36] In the category of vector spaces over a field k, endomorphisms are precisely the linear transformations from a space V to itself. A classic example is scalar multiplication by an element \lambda \in k, which defines the endomorphism \phi_\lambda(v) = \lambda v for all v \in V.^[37] Another representative example is a projection onto a subspace W \subseteq V, which maps vectors in W to themselves and those in a complementary subspace to zero. Such projections are idempotent endomorphisms, satisfying \phi^2 = \phi.^[38] Endomorphisms exhibit notable properties within their rings, including idempotents and nilpotents. An idempotent in \operatorname{End}(S) is an element \phi such that \phi^2 = \phi, generalizing projections in vector spaces to broader structures like modules.^[39] In contrast, a nilpotent endomorphism \phi satisfies \phi^r = 0 for some positive integer r, meaning iterated application eventually yields the zero map; this occurs, for instance, in nilpotent linear operators on finite-dimensional spaces.^[39] These properties play key roles in decomposing structures and analyzing stability. Endomorphisms find applications in dynamical systems, where an endomorphism \phi: X \to X on a space X generates dynamics via iteration, with orbits \{\phi^n(x)\}_{n \geq 0} modeling evolution.^[40] For example, toral endomorphisms induced by integer matrices on the n-torus provide models for chaotic behavior when the matrix has eigenvalues outside the unit circle.^[40] This framework extends to studying invariant measures and ergodicity in non-invertible settings.^[41]

Monomorphisms and Epimorphisms

In category theory, a monomorphism is a morphism \phi: A \to B that is left-cancellative, meaning that if \phi \circ \psi = \phi \circ \psi' for morphisms \psi, \psi': C \to A, then \psi = \psi'.^[42] This property generalizes the notion of an injective function from the category of sets, where monomorphisms coincide exactly with injections, but in more general categories, monomorphisms may not be injective in the underlying sets.^[43] For instance, in the category of abelian groups, the inclusion map \mathbb{Z} \hookrightarrow \mathbb{Q} is a monomorphism because it is injective and satisfies the left-cancellation property. Dually, an epimorphism is a morphism \phi: A \to B that is right-cancellative, meaning that if \psi \circ \phi = \psi' \circ \phi for morphisms \psi, \psi': B \to C, then \psi = \psi'.^[44] In the category of sets, epimorphisms are precisely the surjective functions, but this equivalence does not hold in all categories; for example, in the category of rings, epimorphisms need not be surjective.^[43] A classic illustration is the inclusion \mathbb{Z} \hookrightarrow \mathbb{Q} in the category of rings (with unity), which is an epimorphism: for any two ring homomorphisms f, g: \mathbb{Q} \to R agreeing on \mathbb{Z}, one can show f = g because every rational can be expressed as a quotient of integers, forcing agreement via the universal property.^[45] In contrast, the evaluation map \mathbb{R} \to \mathbb{R} sending a polynomial to its value at $0$ is an epimorphism (and surjective) in the category of rings, but it highlights how projections often behave as epimorphisms in algebraic settings.^[3] While monomorphisms and epimorphisms are one-sided notions, a morphism that is both is an isomorphism in many categories, such as sets or groups, though counterexamples exist in more complex structures like rings.^[46] These concepts emphasize the abstract cancellativity over pointwise properties like injectivity or surjectivity, providing a framework for understanding structure-preserving maps beyond concrete set-theoretic behavior.^[43]

Structural Elements

Kernel

In the context of a homomorphism \phi: S \to T between algebraic structures equipped with an identity element e_T in T, the kernel of \phi, denoted \ker(\phi), is defined as the preimage \ker(\phi) = \{ s \in S \mid \phi(s) = e_T \}.^[47] This set quantifies the extent to which \phi fails to preserve distinct elements of S, as it consists precisely of those elements mapped to the identity, thereby capturing the "loss of injectivity" or structural information in the mapping.^[47] In specific algebraic settings, \ker(\phi) inherits additional structure: for group homomorphisms, it forms a normal subgroup of S; for ring homomorphisms, it is an ideal of S.^[9]/16%3A_Rings/16.05%3A_Ring_Homomorphisms_and_Ideals) The kernel induces a natural congruence relation on S, defined by s \sim t if and only if \phi(s) = \phi(t), or equivalently (in group cases) if s^{-1}t \in \ker(\phi).^[47] This equivalence relation partitions S into cosets, enabling homomorphisms to factor through the quotient structure S / \ker(\phi), where the canonical projection \pi: S \to S / \ker(\phi) followed by an induced map yields \phi.^[48] Specifically, \phi factors as \phi = \overline{\phi} \circ \pi, with \overline{\phi} injective on the quotient, highlighting how the kernel encodes the non-injective part of the homomorphism.^[48] Categorically, the kernel satisfies a universal property: the inclusion morphism k: \ker(\phi) \to S is a monomorphism such that \phi \circ k is the zero morphism (to the terminal object), and for any morphism m: M \to S with \phi \circ m zero, there exists a unique u: M \to \ker(\phi) making the diagram commute, i.e., m = k \circ u.^[49] This property is realized via a pullback diagram in categories with zero morphisms and pullbacks:

\begin{CD} \ker(\phi) @>>> 0 \\ @VVV @VVV \\ S @>{\phi}>> T \end{CD}

where the square is a pullback, ensuring \ker(\phi) is the universal subobject of S annihilated by \phi.^[49] A concrete example arises in the projection homomorphism \pi: \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} given by \pi(k) = k \mod n, whose kernel is \ker(\pi) = n\mathbb{Z}, the subgroup of multiples of n.^[47] This illustrates how the kernel identifies the elements "collapsed" to the identity in the quotient, measuring the structural simplification from \mathbb{Z} to the finite cyclic group.^[47]

Image

In the context of a homomorphism \phi: S \to T between algebraic structures, the image of \phi, denoted \operatorname{im}(\phi), is defined as the set \{\phi(s) \mid s \in S\}. This set forms a substructure of T, such as a subgroup if \phi is a group homomorphism, or a subring if \phi is a ring homomorphism, inheriting the operations from T.^[4] A key property of the image arises from the first isomorphism theorem, which states that the canonical projection map from the quotient S / \ker(\phi) to \operatorname{im}(\phi) is an isomorphism of structures. This theorem establishes that the image is structurally equivalent to the domain modulo its kernel, providing a fundamental link between homomorphisms, quotients, and substructures.^[50] In abelian categories, the cokernel of \phi, denoted \operatorname{coker}(\phi), is the quotient object T / \operatorname{im}(\phi), dual to the kernel construction. This duality highlights the image's role in completing the homological description of \phi, where \operatorname{im}(\phi) serves as the kernel of the canonical map to the cokernel.^[51] For example, consider the inclusion homomorphism \iota: \mathbb{Z} \to \mathbb{Q} defined by \iota(n) = n/1 for n \in \mathbb{Z}. The image \operatorname{im}(\iota) = \mathbb{Z} is the subring of integers within \mathbb{Q}, which algebraically generates \mathbb{Q} as a field when inverted, though it remains a proper substructure./07%3A_Rings_I/7.02%3A_Ring_Homomorphisms)

Applications in Structures

Relational Structures

In relational structures, a homomorphism is a function f: A \to B between two structures A and B sharing the same relational signature, such that for every k-ary relation R in the signature and every tuple (a_1, \dots, a_k) \in R^A, it holds that (f(a_1), \dots, f(a_k)) \in R^B. This preservation ensures that the mapping respects the relational constraints defining the structures, extending the notion of structure-preserving maps from purely algebraic settings to those emphasizing relations over operations.^[52] Examples of relational homomorphisms include order homomorphisms between partially ordered sets (posets), where the binary relation is the order \leq, and f is a monotone map satisfying x \leq y implies f(x) \leq f(y).^[53] In lattices, viewed as posets with an order relation, order homomorphisms similarly take the form of monotone maps that preserve the partial order.^[52] Another instance arises with binary relations, such as adjacency in simple structures, where the homomorphism preserves the relation's membership for pairs.^[54] Relational homomorphisms unify and generalize traditional algebraic homomorphisms by reducing to standard definitions for structures like lattices when relations correspond to operations.^[52] They preserve relational satisfaction for atomic formulas, thereby inducing compatible mappings on any algebraic structures derived from the relations, such as semilattice operations in posets.^[54] In model theory, injective relational homomorphisms serve as embeddings, which not only preserve but also reflect relation membership, ensuring the substructure induced in the codomain matches the original. These embeddings play a key role in constructing extensions and analyzing elementary equivalence between models.

Graph Structures

In graph theory, a homomorphism from a graph G = (V(G), E(G)) to a graph H = (V(H), E(H)) is a mapping f: V(G) \to V(H) such that whenever (u, v) \in E(G), it follows that (f(u), f(v)) \in E(H).^[55] This preservation of adjacency ensures that the structure of connections in G is respected in H, though non-adjacent vertices in G may map to adjacent ones in H.^[56] Unlike isomorphisms, graph homomorphisms need not be bijective or preserve non-edges, allowing for contractions of structure.^[55] A prominent example of graph homomorphisms arises in graph coloring, where a proper k-coloring of G corresponds exactly to a homomorphism from G to the complete graph K_k on k vertices.^[55] In this mapping, vertices of G are assigned colors (vertices of K_k), and the edge preservation condition ensures that adjacent vertices in G map to distinct, adjacent vertices in K_k.^[56] Another example involves retractions, which are homomorphisms from G to an induced subgraph H of G that fix every vertex of H pointwise.^[57] Key properties of graph homomorphisms include the formation of homomorphic images and cores. The homomorphic image of G under a surjective homomorphism f to H is H itself, representing a quotient of G where equivalent vertices (under the kernel of f) are collapsed while retaining edge relations.^[56] A core of a graph is a minimal graph in its homomorphic equivalence class—meaning it admits no homomorphism to any proper subgraph and is thus non-retractable to a smaller graph—unique up to isomorphism and serving as the simplest representative of graphs with identical homomorphism behavior.^[58] Graph homomorphisms find significant application in constraint satisfaction problems (CSPs), where determining the existence of a homomorphism from an instance graph G to a template graph H solves problems like k-coloring (when H = K_k) by verifying satisfiability of edge constraints.^[59] This framework, introduced in seminal work linking graph homomorphisms to general CSPs, enables complexity classifications: for example, CSPs reducible to graph homomorphism are polynomial-time solvable if H is bipartite and NP-complete otherwise in many cases.^[60]

Formal Language Theory

In formal language theory, a homomorphism is a structure-preserving map between the free monoids generated by finite alphabets \Sigma and \Gamma. Formally, it is a function h: \Sigma^* \to \Gamma^* satisfying h(\epsilon) = \epsilon and h(xy) = h(x)h(y) for all words x, y \in \Sigma^*, where \epsilon denotes the empty word.^[61] This mapping is uniquely determined by its action on individual letters, specifying a word h(a) \in \Gamma^* for each a \in \Sigma and extending by concatenation.^[61] Homomorphisms are distinguished as erasing or non-erasing based on whether they permit mapping letters to the empty word. An erasing homomorphism allows h(a) = \epsilon for some a \in \Sigma, effectively deleting symbols; for instance, over \Sigma = \{a, b\}, the mapping h(a) = \epsilon and h(b) = b transforms the word abab to bb by removing all a's.^[61] A non-erasing homomorphism requires h(a) \neq \epsilon for all a \in \Sigma, ensuring no deletions occur; an example is h(a) = ab and h(b) = ba over \{a, b\}, which substitutes each letter with a non-empty word while preserving sequential structure, as seen in applications like rule substitutions in phrase-structure grammars.^[61] These coding functions model operations such as symbol erasure or replacement in language processing and generation.^[61] The kernel of a homomorphism h is the equivalence relation \equiv_h on \Sigma^* defined by u \equiv_h v if and only if h(u) = h(v), partitioning words by their images under h.^[61] In relation to codes, when the kernel aligns with a code (a prefix-free or uniquely decipherable set), the homomorphism supports finite maximal ambiguity in decoding, meaning any image word has at most a bounded number of preimages, distinct from the algebraic kernel in structural elements.^[61] Homomorphisms are essential in automata theory for their preservation properties on language classes. If L \subseteq \Sigma^* is a regular language, then the direct image h(L) = \{ h(w) \mid w \in L \} is regular, as finite automata can be adapted by relabeling transitions according to h.^[61] Similarly, the inverse image h^{-1}(M) = \{ w \in \Sigma^* \mid h(w) \in M \} is regular for any language M \subseteq \Gamma^*, allowing regular languages (also called rational languages) to be mapped while maintaining recognizability by finite automata.^[61] These properties enable reductions in language analysis, such as proving regularity via simplified encodings.^[61]

References

[1]
Abstract Algebra: Homomorphisms - UTSA
Jan 9, 2022 · In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or ...
[2]
[PDF] homomorphisms.pdf - Keith Conrad
In linear algebra this kernel is known as the null space of A. Here is a kernel of a homomorphism that is related to periodicity of the trigonometric functions.
[3]
[PDF] Abstract Algebra I - Lecture 19 - Michigan State University
Oct 26, 2015 · Definition. Let R and S be rings. A function f : R → S is called a homomorphism if for every a,b ∈ R, • f(a + b) = f(a) + f(b), and • f(ab) = f ...
[4]
[PDF] Lecture 4.1: Homomorphisms and isomorphisms
However, we've never really spelled out the details about what this means. We will study a special type of function between groups, called a homomorphism.
[5]
homomorphism in nLab
Aug 19, 2025 · A ring homomorphism is a function between rings that is a homomorphism for both the additive group and the multiplicative monoid. Traditional ...Definitions · Traditional (magmas... · Identity-preserving (monoids... · General
[6]
category in nLab
Sep 19, 2025 · A category consists of a collection of objects and a collection of morphisms. Every morphism has a source object and a target object.Category Theory · Enriched category · 2-Category · Dagger category
[7]
Group Homomorphism -- from Wolfram MathWorld
A group homomorphism is a map between two groups such that the group operation is preserved: for all , where the product on the left-hand side is in and on the ...
[8]
[PDF] 18.703 Modern Algebra, Homomorphisms and kernels
Here are some elementary properties of homomorphisms. Lemma 8.2. Let φ: G -→ H be a homomorphism. (1) φ(e) = f, that is, φ maps the identity in G to the ...
[9]
[PDF] G₁ are groups, a mapping a : G→ G₁ is called a homomorphismⓇ if
Theorem 1 shows that a also preserves the identity, inverses, and powers. Theorem 1. Let a: G G₁ be a homomorphism. Then: (1) a(1) = 1. (2) a(g˜¹) = a(g)−¹ ...
[10]
[PDF] combinatorial aspects of partially ordered sets
Poset Isomorphisms and Duality. Definition 1.7.1. A function φ : P → Q from the poset P into the poset Q is isotone (or order-preserving) if x ≤P y ...
[11]
[PDF] 3. A little Category Theory Definition - UCSD Math
The objects are the collection of all groups and the morphisms are homomorphisms. Composition of morphisms is composition of functions. It is not hard to see ...
[12]
[PDF] The First Isomorphism Theorem
Mar 22, 2018 · The First Isomorphism Theorem. The First Isomorphism Theorem helps identify quotient groups as “known” or “familiar” groups. I'll begin by ...
[13]
[PDF] Automatic Continuity of Group Homomorphisms
The paper surveys automatic continuity of homomorphisms between Polish groups, asking when a homomorphism between them is continuous.
[14]
[PDF] Homomorphisms and Factor/Quotient Rings 𝜙(𝑎 + 𝑏) = 𝜙(𝑎) + 𝜙 ...
We saw earlier that 𝜙: ℤ → ℤ𝑛 by 𝜙(𝑚) = 𝑚 (𝑚𝑜𝑑 𝑛) is a ring homomorphism. Ex. Projection homomorphism ... [ℤ] = ℤ × {0}, but (1,1) is unity for ℤ × ℤ. Analogous to ...
[15]
7.2: Ring Homomorphisms - Mathematics LibreTexts
Mar 5, 2022 · We have the inclusion homomorphism ι : Z → Q , which just sets ι ⁡ ( n ) = n . This map clearly preserves both addition and multiplication.
[16]
9.9: The Matrix of a Linear Transformation - Mathematics LibreTexts
Sep 16, 2022 · This concept is explored in this section, where the linear transformation now maps from one arbitrary vector space to another. Let \(T: V \ ...Example \(\PageIndex{1... · Example \(\PageIndex{2... · Example \(\PageIndex{3...
[17]
Category Theory - Stanford Encyclopedia of Philosophy
Dec 6, 1996 · The category Grp with objects groups and morphisms group homomorphisms, i.e. $(1, \times, (-)^{-1})$ homomorphisms. The category Rings with ...General Definitions, Examples... · Philosophical Significance · Bibliography
[18]
order-preserving map - PlanetMath
Mar 22, 2013 · Order-preserving map from a poset L L to a poset M M is a function f f such that. ∀x,y∈L:(x≥y⟹f(x)≥f(y)).Missing: category | Show results with:category
[19]
functor in nLab
Sep 2, 2025 · A functor is a map between categories that sends objects to objects and morphisms to morphisms, preserving composition.Idea · Definition · Properties · Examples
[20]
hom-set in nLab
May 20, 2023 · In a locally small category, the hom-set hom(x, y) is the collection of all morphisms from x to y. In enriched categories, it is an object of V.
[21]
[PDF] MATH 433 Applied Algebra Lecture 30: Isomorphism of groups ...
In other words, an isomorphism is a bijective homomorphism. The group G is said to be isomorphic to H if there exists an isomorphism f : G → H. Notation: G ∼ = ...
[22]
[PDF] Lecture 3: Homomorphisms and Isomorphisms
First, we define a type of map that is compatible with the group structure on both groups. Defnition 3.1. Given groups G and G ′ , a homomorphism between them ...
[23]
[PDF] Homomorphisms and Isomorphisms
A homomorphism ϕ is an isomorphism if the map ϕ is 1-1 and onto. Therefore, two groups are the “same” if there is an isomorphism ϕ ∶ G Ñ G1. The only difference ...Missing: abstract | Show results with:abstract
[24]
[PDF] Lecture 4.6: Automorphisms - Mathematical and Statistical Sciences
Basic concepts. Definition. An automorphism is an isomorphism from a group to itself. The set of all automorphisms of G forms a group, called the automorphism ...
[25]
[PDF] automorphism groups of simple graphs
An isomorphism is a homomorphism that is one-to-one and onto. Definition 1.5. An automorphism of a group G is an isomorphism from G to itself. We denote the set ...
[26]
[PDF] dihedral groups ii - keith conrad
We will characterize dihedral groups in terms of generators and relations, and describe the subgroups of Dn, including the normal subgroups.
[27]
[PDF] 10. Basic properties of rings Lemma 10.1. Let R be a ... - UCSD Math
Example 10.6. Consider the two rings Z and 2Z. These are isomorphic as groups, since the function Z −→ 2Z which sends n −→ 2n,
[28]
[PDF] FIELD AUTOMORPHISMS OF R AND Qp - Keith Conrad
Introduction. An automorphism of a field K is an isomorphism of K with itself: a function f : K → K that is a bijective field homomorphism (additive and ...
[29]
[PDF] group properties and group isomorphism - UCSD Math
May 25, 2025 · Groups posses various properties or features that are preserved in isomorphism. An isomorphism preserves properties like the order of the group,.
[30]
[PDF] §3.4 Isomorphisms - University of South Carolina
Any group isomorphism preserves general products, the identity and inverses. Shaoyun Yi. Isomorphisms. Summer 2021. 5 / 16. Page 6 ...
[31]
[PDF] Isomorphisms Math 130 Linear Algebra
Whereas isomorphisms are bijections that pre- serve the algebraic structure, homomorphisms are simply functions that preserve the algebraic struc- ture. In the ...
[32]
http://math.clarku.edu/~ma130/isomorphism.pdf
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[PDF] Abstract Algebra. Math 6310. Bertram/Utah 2022-23. Rings ...
Definition. A commutative ring with 1 (R, +, ·) is a set R with ... Recall that the ring of k-linear endomorphisms: Mn×n(k) = Endvs(kn) is the ...
[34]
[PDF] Rings and Modules
Let A be an abelian group, End(A) = Hom(A, A). (End(A), +, ◦) is the endomorphism ring of A. Fact. Every ring R is a subring of ...
[35]
[PDF] 7 Endomorphism rings
Feb 26, 2015 · Definition 7.5. Let E/k be an elliptic curve. The endomorphism ring of E is the additive group End(E) := hom(E,E) with multiplication defined ...
[36]
[PDF] Endomorphisms, Automorphisms, and Change of Basis
The set of all endomorphisms of V will be denoted by L (V;V ). A vector space isomorphism that maps V to itself is called an automorphism of V .
[37]
[PDF] Clifford Algebras and Bott Periodicity
Aug 29, 2023 · An idempotent endomorphism is also called a projection. Notice that in the category of vector spaces, an idempotent endomorphism is just a ...
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[PDF] 2.3 Nilpotent endomorphisms
Definition 2.3. 1. An endomorphism f ∈ EndC(V ) is called nilpotent if there exists r ∈ N such that f r = 0EndC(V ), so that f r (v)=0V , for every v ∈ V .
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[PDF] Dynamics of linear maps (continued). Toral endomorphisms.
Let LA denote a toral endomorphism induced by the linear map. L(x) = Ax, x ∈ Rn. The map LA is a toral automorphism if it is invertible. Proposition The ...
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[PDF] Endomorphisms of the shift dynamical system, discrete derivatives ...
Mar 24, 2009 · We prove that D, V ◦D, S, and V ◦S are conjugate to S and are the only continuous endomorphisms of S whose parity vector function is solenoidal.
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[PDF] Basic Concepts in category theory
Definition 1.5. We say a morphism f : A → B is a monomorphism if fg = fh ⇒ g = h for all g, h : C → A.
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[PDF] Notes on Category Theory - UT Math
Feb 28, 2018 · Definition 6.2. A morphism f : X → Y is a monomorphism if it is left cancellative. That is, for all g, h : Z → X,.
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[PDF] CATEGORY THEORY Contents 1. Definitions and Examples 2 2 ...
Oct 9, 2006 · Definition 2.7. We say a morphism f : A → B is a monomorphism if fg = fh ⇒ g = h for all g, h : C → A.
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[PDF] Math 762 Spring 2016 Homework 1 Drew Armstrong
Since every element of Q can be written in the form i(p)i(q)−1 for some p, q ∈ Z this implies that β1 = β2 and hence i : Z → Q is an epimorphism. D.<|control11|><|separator|>
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[PDF] Chapter 1 Category Theory
Proof: Let f : X → Y be a monomorphism. Given j : A → X such that fj = 0, since we also have f0 = 0 the definition of monomorphism gives j = 0, ...
[46]
https://www.math.toronto.edu/selick/mat1352/1350notes.pdf
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[PDF] Kernels and quotients - Purdue Math
If H ✓ G is a normal subgroup, then G/H becomes a group with respect to the product defined above. The map p(g) = gH is a homomorphism with kernel H. Proof. By ...Missing: factor | Show results with:factor
[48]
kernel in nLab
Apr 12, 2025 · In any category enriched over pointed sets, the kernel of a morphism f : c → d f:c\to d is the universal morphism k : a → c k:a\to c such that f ...
[49]
Group Theory - The Isomorphism Theorems
First Isomorphism Theorem: Let be a group homomorphism. Let E be the subset of G that is mapped to the identity of G ′ . E is called the kernel of the map φ.
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[PDF] Abelian Categories - Purdue Math
Jan 28, 2024 · The cokernel of f : M → N in this category is the quotient. N/f(M). However, if f(M) is not closed, the map from coker(ker f → M) = M/ ker f to ...
[51]
Relational homomorphisms - ScienceDirect.com
It is argued that a promising way to unify and generalize homomorphisms is to follow a definition from universal algebra. This definition reduces to the usual ...
[52]
[PDF] Lecture Notes: Introduction to Categorical Logic - andrew.cmu.ed
Jun 1, 2009 · Graph is the category of (directed) graphs an graph homomorphisms. • Poset is the category of posets and monotone maps. Such categories of ...
[53]
Compositional Homomorphisms of Relational Structures
The paper attempts a systematic study of homomorphisms of relational structures. Such structures are modeled as multialgebras (i.e., relation is represented ...<|control11|><|separator|>
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[PDF] Graph homomorphisms
A homomor- phism from the graph G to the complete graph Kr (with vertices numbered. 1,2,...,r) is exactly the same as an r-colouring of G (where the colour of a.
[55]
Graphs and Homomorphisms - Pavol Hell; Jaroslav Nešetřil
The text contains exercises of varying difficulty, intended to support readers' learning and further development. Show more. Cover. Graphs and Homomorphisms.
[56]
[PDF] Graphs and Algorithms in Constraint Satisfaction - LIX
Nov 26, 2008 · Show that f : V (G) → V (H) is a homomorphism from G to H if and only if f is a homomorphism from the undirected graph of G to the undirected ...<|control11|><|separator|>
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The core of a graph - ScienceDirect.com
We investigate some homomorphism properties of cores and conclude that it is NP-complete to decide whether or not a graph is its own core. (A similar conclusion ...Missing: retractable | Show results with:retractable
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[PDF] From Graph Colouring to Constraint Satisfaction:
The Constraint Satisfaction Problem for a fixed a structure H: CSP(H) • Given a structure G (same vocabulary as H) • Is there a homomorphism of G to H ?
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The complexity of homomorphism and constraint satisfaction ...
The paper classifies homomorphism problems for graphs, relating them to constraint satisfaction problems and database theory, and links them to bounded tree ...
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[PDF] 7 Morphisms Tero Harju and Juhani Karhum¿aki Department of ...
in fact, h is a prefix code and g is a suffi x code. This example ... kernel of the morphism h. Therefore h is a solution, if one obtains equal words ...