1 Semigroups and Algebraic Foundations

1.1 Semigroups: definitions and basic examples

A semigroup is a set \(S\) equipped with an associative binary operation \(\cdot : S\times S \to S\). Associativity means \((ab)c=a(bc)\) for all \(a,b,c\in S\). In contrast to groups, elements need not have inverses, and there may be no identity element.

Typical examples include:

  • Multiplication of natural numbers: \((\mathbb{N},\times)\) is a semigroup since multiplication is associative.
  • String concatenation: the set of all finite words over an alphabet forms a semigroup under concatenation.
  • Transformation semigroups: a set of functions on a set, closed under composition, is a semigroup.
  • Matrix semigroups: a set of matrices closed under multiplication forms a semigroup (often without requiring inverses).

1.2 Semigroup operations and associative multiplication

The defining feature of a semigroup is that its operation can be composed repeatedly without ambiguity. Algebraic constructions built from \(S\) rely on this associativity to ensure that multiplication in derived algebraic structures is also associative.

When a semigroup is given concretely (e.g., as transformations), the semigroup operation is typically explicit—such as function composition—so that associativity follows from the associativity of composition.

1.3 Identities, zero elements, and special elements (idempotents, units)

Semigroups may include distinguished elements:

  • An identity element \(e\) satisfies \(ea=a=ae\) for all \(a\in S\). A semigroup with an identity is sometimes called a monoid.
  • A zero element \(0\) satisfies \(a0=0a=0\) for all \(a\in S\).
  • An idempotent \(e\) satisfies \(e^2=e\). Idempotents are central in semigroup theory because they strongly constrain the structure of Green’s relations and representation behavior.
  • A unit-like element is not standard terminology for semigroups, but one often discusses elements that behave invertibly inside particular substructures (e.g., within maximal subgroups attached to idempotents). In general semigroups, true global inverses may be absent.

These notions influence the corresponding semigroup algebra: idempotents often correspond to special elements used to construct modules and decompositions, while zeros affect annihilation phenomena.

2 Construction of a Semigroup Algebra

2.1 Base ring or field choices

Given a semigroup \(S\) and a commutative base ring \(k\) (often a field), the semigroup algebra is built so that coefficients come from \(k\). The choice of \(k\) affects representation-theoretic outcomes: for instance, semisimplicity and decomposition properties depend on the characteristic of \(k\) and on whether \(k\) is algebraically closed.

In many references, the algebra is written \(kS\) or \(k\langle S\rangle\) and is defined when \(S\) is finite or infinite alike. For infinite \(S\), elements are typically still finite linear combinations of basis symbols indexed by elements of \(S\).

2.2 Definition via formal linear combinations

The construction begins with a free \(k\)-module with basis \(S\).

Concretely, an element of the semigroup algebra is a finite sum \[ \sum_{i} a_i s_i, \] where \(a_i\in k\) and \(s_i\in S\) are distinct basis elements. This viewpoint treats elements of \(S\) as formal basis vectors.

2.2.1 Multiplication rule and bilinear extension

Multiplication on basis elements is induced by the semigroup operation: \[ (s)(t) = st \quad \text{for } s,t\in S. \] This is extended to all of \(kS\) by \(k\)-bilinearity: for \(x=\sum_i a_i s_i\) and \(y=\sum_j b_j t_j\), \[ xy=\sum_{i,j} a_i b_j (s_i t_j). \] Associativity of semigroup multiplication guarantees associativity in \(kS\), since the product is defined by bilinear extension of an associative operation.

2.2.2 Identity and augmentation when applicable

If \(S\) is a monoid with identity \(e\), then \(e\) becomes the multiplicative identity of \(kS\). If \(S\) has a zero element \(0\), then the basis element \(0\) acts as a multiplicative annihilator in \(kS\).

In cases where one uses a monoid with an identity, there is often a related augmentation perspective: maps that “sum coefficients” or collapse the basis according to the monoid’s structure can be used to separate the identity component from the rest. For semigroup (non-monoid) settings, analogous bookkeeping can be adapted, but the canonical presence of an identity is not guaranteed.

2.3 Natural embeddings and canonical maps

Each semigroup element corresponds to a basis element in the semigroup algebra. Thus there is a canonical \(k\)-linear embedding \[ S \hookrightarrow kS,\quad s \mapsto s \] (where the right-hand side denotes the corresponding basis vector).

Additionally, semigroup homomorphisms induce algebra homomorphisms (discussed in detail later). Quotients and substructures of semigroups often lead to induced algebra homomorphisms via the universal property implicit in the bilinear extension.

2.4 Grading and filtration viewpoints (when relevant)

If a semigroup comes with a natural grading or filtration (for example, by length in a word semigroup or by rank in a transformation semigroup), the semigroup algebra can inherit a grading by collecting basis elements of the same “degree.” This can organize computations and can be used to analyze ideals through graded methods.

Even when no strict grading exists, filtrations can be useful: one may define an increasing chain of subspaces spanned by elements satisfying bounds from the underlying combinatorial parameter, and then study how multiplication interacts with the filtration.

3 Ideal Structure and Correspondence with Semigroup Ideals

3.1 Two-sided ideals in semigroup algebras

The semigroup algebra \(kS\) is a ring (if \(k\) is commutative), and it has its own two-sided ideals. These ideals reflect how subsets of \(S\) interact under multiplication.

A fundamental principle is that many algebraic ideals arise from semigroup-level subsets. However, the correspondence is not always one-to-one: algebraic ideals can mix basis elements in coefficient-dependent ways, especially when the base ring has zero divisors or when characteristic constraints introduce additional relations.

3.2 Semigroup ideals induced to algebra ideals

A two-sided ideal of \(S\) is a subset \(I\subseteq S\) such that \(SIS\subseteq I\). Given such \(I\), one can form the \(k\)-submodule \[ kI=\left\{\sum a_i i_i : i_i\in I\right\}\subseteq kS. \] Because multiplication of basis elements respects the semigroup product, one finds that \(kI\) is a two-sided ideal of \(kS\).

Conversely, under suitable hypotheses, every two-sided ideal of \(kS\) may be closely related to a semigroup ideal, but the precise relationship depends on \(k\) and on how \(S\) sits inside \(kS\). In general, semigroup ideals generate algebraic ideals in an inclusion-preserving way: larger semigroup ideals produce larger algebra ideals.

3.3 Green’s relations and algebraic consequences

Green’s relations partition a semigroup according to generated principal ideals. They refine the semigroup’s internal structure and provide a systematic language for describing regularity, invertibility inside substructures, and the behavior of idempotents.

At the algebra level, these relations influence:

  • the shape of ideals and their quotients,
  • decomposition patterns in representation theory,
  • and the organization of certain module categories.

In many settings, idempotent-based data (coming from Green’s theory) yields explicit algebra elements or module idempotents that control how representations break apart.

Semigroup theory distinguishes “regular” elements—those that satisfy a form of inner inverse condition—from more general elements. Likewise, inverse semigroups have strongly constrained idempotent structure.

The semigroup algebra often mirrors this split: regular/inverse components can correspond to well-behaved subalgebras or quotients, while the non-regular part tends to contribute to radicals or to non-semisimple behavior. These are conceptual links rather than a universal theorem for all \(S\), but the guiding idea is that semigroup regularity patterns translate into algebraic regularity or failure thereof.

3.4 J-classes, principal factors, and algebra modules

A \(J\)-class in a semigroup describes elements related under the two-sided ideal generated by them: \(a\) and \(b\) lie in the same \(J\)-class if they generate the same principal two-sided ideal. The set of \(J\)-classes partially orders the semigroup via inclusion relations among generated ideals.

In the semigroup algebra, this ordering can help organize the structure of ideals and modules:

  • Principal factors—quotients built from consecutive ideal layers—often correspond to building blocks in the module category.
  • Simple modules and projective covers frequently arise from these factors, particularly when one analyzes the algebra with idempotent and \(J\)-class techniques.

4 Representation Theory of Semigroup Algebras

4.1 Modules over semigroup algebras

A left module over \(kS\) is a \(k\)-module \(M\) with an action map \(kS\times M\to M\) compatible with multiplication. Since the basis of \(kS\) is indexed by \(S\), specifying the module is equivalent to giving a \(k\)-linear representation of \(S\) on \(M\): \[ s\cdot m \] for \(s\in S\), such that \((st)\cdot m = s\cdot (t\cdot m)\). The module axioms ensure that this action is associative in the sense inherited from \(S\).

4.2 Simple modules and their origins from semigroup structure

Simple modules are nonzero modules with no proper nontrivial submodules. Their classification in the semigroup algebra setting typically depends on:

  • idempotents and their associated local structures,
  • Green’s relations that determine how elements act “locally,”
  • and radical behavior that affects whether semisimplicity holds.

A recurring theme is that simple modules often attach to specific semigroup components (commonly determined by idempotents and \(J\)-classes). Representation theory thus becomes a translation of semigroup combinatorics into linear algebra.

4.3 Semigroup representations as algebra homomorphisms

A representation of a semigroup on a \(k\)-vector space \(V\) (or \(k\)-module) corresponds to an algebra homomorphism \[ kS \to \operatorname{End}_k(V). \] Indeed, sending \(s\in S\) to the corresponding endomorphism (the linear action of \(s\)) respects multiplication because of the semigroup identity \((st)\cdot v=s\cdot(t\cdot v)\). The universal construction of \(kS\) ensures that the linearization process produces a unique algebra homomorphism.

4.4 Characteristic idempotents and module decomposition

When semigroups contain idempotents, the semigroup algebra often exhibits idempotent-like elements (sometimes constructed from the original semigroup idempotents and additional local data). Such idempotents can serve as “projectors” onto submodules, enabling decomposition strategies.

In semisimple situations, idempotent projectors can split modules directly. In more general (non-semisimple) cases, idempotents still guide the structure of indecomposable modules and determine the shape of filtration series, even if complete splitting does not occur.

5 Special Classes of Semigroups

5.1 Commutative semigroups and commutative semigroup algebras

If \(S\) is commutative, then the semigroup algebra \(kS\) is commutative as a ring because multiplication of basis elements is induced by the commutative operation in \(S\). Commutative semigroup algebras connect to commutative algebra methods: spectra, prime ideals, and decomposition into local components can sometimes be applied, especially when \(S\) is finite and \(k\) is a field.

5.2 Finite semigroups and finite-dimensional algebras

When \(S\) is finite, \(kS\) is finite-dimensional over \(k\). This finite-dimensionality makes many questions tractable:

  • the module category behaves more like the usual finite-dimensional associative algebra case,
  • radicals and semisimple quotients can be studied using standard tools,
  • and computational approaches like determining the multiplication table become feasible.

5.3 Cancellative, inverse, and regular semigroups (structural remarks)

Different structural constraints on \(S\) impose additional algebraic restrictions on \(kS\):

  • In cancellative settings, fewer identifications occur in the semigroup, often simplifying certain ideal and representation patterns.
  • Inverse semigroups have a robust idempotent theory; their semigroup algebras can be analyzed via local group behavior attached to idempotents.
  • Regular semigroups tend to yield algebraic structures with better-controlled module behavior, though non-semisimplicity may still occur depending on \(k\).

These remarks are often used to predict how strongly semigroup regularity will constrain the algebra’s radical and representation-theoretic complexity.

5.4 Nilpotent and locally nilpotent behavior in the algebra

If the semigroup has “nilpotent-like” behavior (for instance, products of sufficiently long sequences eventually hit a zero element or vanish in a quotient), the semigroup algebra inherits related features:

  • certain ideals become nil (or nilpotent),
  • powers of the augmentation ideal may collapse,
  • and representation theory can reflect that actions eventually become trivial on many modules.

Exact equivalences depend on definitions and on whether one studies the semigroup itself or the algebra with additional quotients, but the general phenomenon is that semigroup nilpotence translates into algebraic nilpotence constraints.

6 Structural Properties of Semigroup Algebras

6.1 Rings of semigroup algebras: overview of key properties

Semigroup algebras are associative \(k\)-algebras whose basis elements reproduce the semigroup multiplication. Basic ring-theoretic invariants—ideals, radicals, quotient rings, and module decompositions—can be studied in terms of semigroup structure.

For finite semigroups over fields, \(kS\) becomes a standard finite-dimensional associative algebra, so classical ring-theoretic concepts apply directly.

6.2 Jacobson radical considerations (high-level)

The Jacobson radical measures the “non-semisimple” part of an algebra. In semigroup algebras, radical structure is influenced by:

  • the presence of nilpotent elements or collapsing products in \(S\),
  • the arrangement of \(J\)-classes and principal factors,
  • and characteristic-dependent phenomena.

While explicit formulas can be complicated, a high-level viewpoint is that parts of the semigroup producing degeneracies (such as elements that are far from invertible behavior) contribute to radical layers in \(kS\).

6.3 Semisimplicity criteria (general discussion)

A finite-dimensional algebra is semisimple if it is a direct sum of matrix algebras over division rings; over an algebraically closed field, this becomes a direct sum of full matrix algebras over the field.

For semigroup algebras, semisimplicity is rare and depends heavily on both the semigroup’s structural regularity and on the characteristic of \(k\). If the semigroup contains complicated nilpotent behavior, semisimplicity typically fails. Conversely, when the semigroup is close to a group-like structure and the base field is compatible, semisimplicity may occur in constrained cases.

Semiprimeness excludes nonzero nilpotent ideals. In the semigroup algebra context, semiprimeness relates to whether the semigroup has nilpotent ideal behavior that would create nilpotent algebra ideals.

Conceptually, semiprime properties in \(kS\) correspond to semigroup constraints that prevent collapse under repeated multiplication, especially in the two-sided ideal structure. Because the semigroup algebra linearizes \(S\), even subtle semigroup phenomena can yield algebraic nilpotent behavior.

7 Homomorphisms, Functoriality, and Morphisms

7.1 Induced algebra maps from semigroup maps

If \(\varphi:S\to T\) is a semigroup homomorphism, there is an induced \(k\)-algebra homomorphism \[ kS \to kT \] defined on basis elements by \(s\mapsto \varphi(s)\) and extended linearly. This follows because multiplication in \(kS\) corresponds to multiplication in \(S\), and \(\varphi(st)=\varphi(s)\varphi(t)\).

Thus, the semigroup algebra construction behaves functorially: composition of semigroup homomorphisms corresponds to composition of algebra homomorphisms.

7.2 Quotients and subalgebras corresponding to semigroup quotients

If \(S\) is factored by a semigroup congruence (equivalence compatible with multiplication), the induced structure on \(kS\) often yields an algebra quotient. At a practical level, identifying basis elements according to the congruence produces relations in the algebra.

Similarly, subsemigroups can correspond to subalgebras: the \(k\)-span of a subsemigroup is a subalgebra if the subsemigroup is closed under multiplication, which it is by definition.

7.3 Direct products and coproduct-like constructions (when used)

Semigroup products have corresponding algebraic products. For finite products, one typically has correspondences between:

  • the semigroup algebra of a direct product and tensor products of algebras (under standard finiteness and base-ring assumptions),
  • and the ways in which independent semigroup components act on modules.

These relationships allow one to build semigroup algebra examples systematically from simpler pieces.

7.4 Change of base ring/field and scalar extension

If one enlarges the base ring from \(k\) to \(K\), the semigroup algebra can be extended by scalars: \[ K\otimes_k kS \cong KS, \] with \(KS\) defined using \(K\) as the coefficient ring. Scalar extension can change decomposition behavior: modules that were indecomposable over \(k\) may split over \(K\), and representation dimensions remain controlled but multiplicities can change.

8 Computation and Examples

8.1 Small semigroups worked examples

For small finite semigroups, \(kS\) can be written explicitly by listing basis elements and computing products using the multiplication table of \(S\). Ideals can also be found by examining spans of subsets closed under the two-sided action.

Even simple semigroups with a zero element illustrate how annihilation in the semigroup forces certain products to vanish in the algebra, immediately constraining ideal structure.

8.2 Presentation by generators and relations (algebraic perspective)

Although semigroup algebras are defined from a semigroup, one can often present them using generators and relations inherited from the semigroup operation. If \(S\) is described by generators with defining relations (as a semigroup), then \(kS\) becomes a quotient of a free associative algebra by the corresponding relations, linearized over \(k\).

This perspective is useful when the semigroup is not given by a full multiplication table but instead by an abstract specification.

8.3 Determining multiplication tables in the algebra

Once the basis correspondence is set, multiplication in \(kS\) is determined by bilinearity:

  • compute \(s_it_j\) in \(S\),
  • then express the result as the corresponding basis element,
  • and finally combine coefficients.

This procedure yields structure constants for \(kS\) relative to the basis \(S\). For finite \(S\), the outcome is a complete description of the algebra’s multiplication.

8.4 Example: semigroup algebras from transformation semigroups (overview)

A transformation semigroup consists of functions \(X\to X\) closed under composition. Its semigroup algebra has basis elements indexed by transformations, with multiplication given by composition. Modules over such an algebra correspond to linear actions of transformations, which can be interpreted as linearized permutation-like behavior, though without requiring invertibility.

From a computational standpoint, one can often use the rank or image structure of transformations to stratify basis elements and analyze how products shrink or consolidate images, producing concrete ideal and module constraints.

9 Connections and Applications

When a semigroup is actually a group, the semigroup algebra coincides with the usual group algebra. Many representation-theoretic themes—simple modules, character-like data, and decomposition—therefore generalize from groups to semigroups, though with additional complications due to non-invertible behavior.

Conversely, semigroup algebras can be viewed as extensions of group-algebra techniques that accommodate broader multiplicative phenomena.

9.2 Semigroup actions on sets and induced linear representations

A semigroup action on a set \(X\) provides a combinatorial way to build representations: one can consider the \(k\)-module of functions on \(X\) (or the free \(k\)-module on \(X\)) and let each semigroup element act by transporting arguments. This converts a set-theoretic action into a linear representation, which is equivalently a module structure over \(kS\).

This is a standard bridge between semigroup theory and linear algebra, turning combinatorial dynamics into algebraic data.

9.3 Applications in automata theory and formal languages (algebraic angle)

Finite semigroups arise naturally from automata: the transition semigroup of a deterministic automaton is the semigroup generated by its state transition functions. The semigroup algebra then packages the transition behavior into an associative algebra, enabling the use of algebraic methods to study language recognition.

In this context, representation theory can be interpreted as describing how automaton behavior decomposes into linear components, which can inform minimization and structural analysis.

9.4 Use in studying associative algebra structures arising from combinatorics

Combinatorially defined semigroups (often formed from words, relations, tableaux-like operations, or partial symmetries) produce semigroup algebras that serve as concrete associative algebras. Their ideals and modules reflect the combinatorics: principal factors correspond to layers of complexity in the combinatorial object, and idempotents often encode stabilization phenomena.

These constructions provide a tractable way to convert discrete operations into algebraic invariants, supporting both theoretical classification and explicit computation.