1 Basic Definitions and Notation

1.1 Semigroups and Associative Operations

A semigroup is a pair \((S,\cdot)\) where \(S\) is a set and \(\cdot : S\times S \to S\) is a binary operation such that associativity holds: \[ (x\cdot y)\cdot z = x\cdot (y\cdot z)\quad \text{for all }x,y,z\in S. \] No further axioms are required. In particular, a semigroup need not have an identity element (a neutral element for \(\cdot\)) and need not have inverses for its elements.

In notation, the operation is often written by juxtaposition (\(xy\)) or as a symbol depending on context. Powers are defined unambiguously by associativity: \(x^n\) means \(x\) multiplied with itself \(n\) times.

1.2 Examples of Semigroups

1.2.1 Finite semigroups from multiplication tables

Any finite set equipped with an associative operation can be described by a multiplication table. For instance, if \(S=\{a,b,c\}\), then specifying \(ab,ac,ba,\dots\) determines the operation. The structure is a semigroup precisely when the table entries satisfy associativity for every triple of elements.

Such table-based examples are common in classification problems because they allow direct computation of properties like ideals, Green’s relations, and element regularity.

1.2.2 Transformation and function semigroups

Let \(X\) be a set. Consider all functions \(f:X\to X\) under composition. Since function composition is associative, the set of all endofunctions forms a semigroup, typically denoted \(T(X)\). When \(X\) is finite, \(T(X)\) is a finite semigroup whose size is \(X^{X}\).

More generally, a transformation semigroup is any subsemigroup of \(T(X)\), obtained by restricting to transformations generated by chosen functions.

1.2.3 String concatenation semigroups

Let \(A\) be an alphabet and consider all finite strings over \(A\). With concatenation as the binary operation, \[ u\cdot v := uv, \] associativity follows from the associativity of concatenating finite sequences. If the empty string is excluded, one obtains a semigroup; if it is included, one gets a monoid. In semigroup theory, the version without an identity is often used to emphasize the role of associativity alone.

1.3 Subsemigroups and Generated Subsemigroups

1.3.1 Closure under the operation

A subset \(T\subseteq S\) is a subsemigroup of \(S\) if it is closed under the semigroup operation: for all \(x,y\in T\), the product \(xy\) lies in \(T\). Associativity for \(T\) is inherited automatically from \(S\).

1.3.2 Semigroup generated by a set

Given a subset \(A\subseteq S\), the subsemigroup generated by \(A\) is the smallest subsemigroup containing \(A\). Concretely, it consists of all finite products of elements of \(A\) (with arbitrary parenthesization, which is irrelevant due to associativity). This construction provides a controlled way to build larger semigroups from specified “building blocks.”

2 Structural Concepts

2.1 Ideals and Quotients

2.1.1 Left, right, and two-sided ideals

2.1.1.1 Principal ideals and their behavior

An element-based way to describe “absorption” uses ideals. A left ideal \(I\subseteq S\) satisfies \(SI\subseteq I\); a right ideal satisfies \(IS\subseteq I\); and a two-sided ideal satisfies both.

A principal ideal is the ideal generated by one element. For example, the principal left ideal generated by \(a\in S\) is \(Sa=\{xa : x\in S\}\), and the principal right ideal is \(aS=\{ax:x\in S\}\). These sets are always left or right ideals respectively. Their structure helps organize elements by how they produce other elements under multiplication.

2.1.2 Quotient semigroups via congruences

Semigroup “quotients” are formed using congruence relations. A congruence \(\sim\) on \(S\) is an equivalence relation compatible with multiplication: \[ x\sim y,\ u\sim v \implies xu \sim yv. \] Once such a relation is fixed, the set of equivalence classes \(S/\!\sim\) becomes a semigroup by defining \([x]\cdot [y]=[xy]\). This is analogous to quotient constructions in rings or groups, but without assuming inverses.

When an ideal \(I\) is singled out, one can form a Rees quotient, which collapses everything in \(I\) to a distinguished “zero-like” class while keeping the remaining elements separate. Rees quotients are particularly useful for analyzing how ideal structure influences the global behavior of the semigroup, especially in finite cases where iteration of quotient steps can be organized.

2.2 Green’s Relations

2.2.1 Definitions of 𝓛, 𝓡, 𝓙

Green’s relations partition a semigroup into subsets that capture how elements generate the same “principal ideal” patterns.

  • \(a \,\mathcal{L}\, b\) if \(Sa = Sb\) (same principal left ideal).
  • \(a \,\mathcal{R}\, b\) if \(aS = bS\) (same principal right ideal).
  • \(a \,\mathcal{J}\, b\) if \(SaS = SbS\) (same principal two-sided ideal).

These equivalence relations organize elements by their behavior under multiplication from the left, right, or both sides.

2.2.2 Consequences and comparability

Green’s relations come with built-in order structure: for instance, \(a \le_{\mathcal{J}} b\) is often formulated in terms of inclusion of the corresponding two-sided ideals. Such comparisons help distinguish elements that are “lower” or “higher” in how they are produced within the semigroup. In finite semigroups, this can be turned into practical schemes for computing relational classes and for understanding decomposition into simpler components.

2.2.3 𝓗-relation and group substructures

The \(\mathcal{H}\)-relation refines \(\mathcal{L}\) and \(\mathcal{R}\): typically, \[ a\,\mathcal{H}\, b \quad \text{iff} \quad a\,\mathcal{L}\, b \text{ and } a\,\mathcal{R}\, b. \] Within an \(\mathcal{H}\)-class, additional structure may emerge. In regular contexts, \(\mathcal{H}\)-classes can behave like group-like components, making Green’s framework a central tool for linking semigroup theory to group theory without requiring global invertibility.

2.3 Regular and Inverse Semigroups (Overview)

2.3.1 Regular elements and regular semigroups

An element \(a\in S\) is regular if there exists \(b\in S\) such that \[ a = aba. \] A semigroup in which every element is regular is called regular. Regularity captures a controlled form of “near invertibility,” where multiplication can reproduce the element using an auxiliary factor even though true inverses may not exist.

2.3.2 Idempotents and their role

An idempotent is an element \(e\in S\) satisfying \(e^2=e\). Idempotents are important because many structural decompositions are guided by them: they often serve as anchors for ideal behavior, Green’s relations, and regularity. In many semigroups, sets of idempotents carry order-like structures or semilattice-like relationships.

2.3.3 Inverse semigroups as a special class

An inverse semigroup is a semigroup where every element has a unique “inverse” in the sense that for each \(a\) there is an element \(a^{-1}\) such that \[ aa^{-1}a = a,\quad a^{-1}aa^{-1} = a^{-1}. \] Inverse semigroups generalize groups while preserving strong structural regularity. Their idempotents form a commutative substructure, and Green’s relations interact particularly well with the inverse operation.

3 Algebraic Constructions

3.1 Products and Direct Constructions

3.1.1 Direct product of semigroups

Given semigroups \((S,\cdot)\) and \((T,\ast)\), their direct product is \((S\times T,\diamond)\) where \[ (s,t)\diamond (s',t') := (ss',tt'). \] Associativity holds componentwise, making the product semigroup a standard method for combining independent systems.

3.1.2 Subdirect products and embeddings

A subdirect product is a subsemigroup \(U\subseteq S\times T\) whose coordinate projections onto \(S\) and \(T\) are surjective. Such constructions allow one to represent a semigroup as a “simultaneous compatibility” of two factors while retaining information about how they interact. Embeddings into products provide a way to study semigroups through their images in more manageable settings.

3.2 Homomorphisms and Congruences

3.2.1 Semigroup homomorphisms

A function \(\varphi:S\to T\) between semigroups is a homomorphism if \[ \varphi(xy)=\varphi(x)\varphi(y) \] for all \(x,y\in S\). Homomorphisms preserve the operational structure, so they map subsemigroups to subsemigroups (not necessarily surjectively), and they send associative composites to associative composites.

3.2.2 Congruence relations

Congruences are the equivalence relations compatible with multiplication that allow quotient constructions. Given a homomorphism \(\varphi\), its kernel congruence can be defined by \[ x \equiv y \quad \text{iff} \quad \varphi(x)=\varphi(y). \] This links morphisms to quotients and provides a common theme: structural simplification corresponds to identifying elements that behave the same under all products as seen through \(\varphi\).

3.2.3 The congruence lattice viewpoint

All congruences on a fixed semigroup partially order by refinement. Under inclusion, they form a lattice (with meet corresponding to intersection-like behavior and join corresponding to the smallest congruence containing two given ones). Viewing semigroups through this congruence lattice helps classify them via how many distinct quotient behaviors they admit.

3.3 Free Semigroups and Presentations

3.3.1 Words and formal concatenation

A word over a set \(X\) of symbols is a finite sequence of elements from \(X\). Concatenation of words is associative, producing the basis for a free semigroup: there is no imposed identification beyond literal concatenation.

3.3.2 Free semigroups on a set of generators

The free semigroup on \(X\), often denoted \(X^+\), consists of all nonempty words over \(X\). It has a universal property: any function from \(X\) into a semigroup \(S\) extends uniquely to a homomorphism from \(X^+\) to \(S\). This property makes free semigroups the starting point for defining semigroups by generators and relations.

3.3.3 Presentations and quotient by relations

A presentation of a semigroup describes it as a quotient of a free semigroup by relations among words. One specifies a set of generators \(X\) and a set of defining equations \(u=v\). The resulting semigroup is the free semigroup modulo the smallest congruence that makes all stated relations hold. Presentations are a central language for describing families of semigroups and for constructing examples with controlled algebraic behavior.

4 Special Classes and Key Properties

4.1 Commutative Semigroups

4.1.1 Commutativity vs. associativity

A semigroup is commutative if \(xy=yx\) for all \(x,y\). Associativity is always present by definition, but commutativity adds a strong extra constraint that collapses many distinct behaviors. When commutativity holds, the study of powers, ideals, and idempotents often becomes more structured and easier to classify.

4.1.2 Examples and typical behaviors

Commutative semigroups arise naturally from operations like multiset union or addition without inverses, depending on the context. In many commutative settings, principal ideals tend to relate more closely to divisibility-like relations, and Green’s relations often merge or simplify because left and right multiplication behave similarly.

4.2.1 Bands and semilattices

A band is a semigroup in which every element is idempotent. This means \(x^2=x\) for all \(x\). Bands encode a “projection-like” behavior: multiplying an element by itself never changes it.

A semilattice is a commutative band in which multiplication is associative and idempotent, and commutativity ensures a natural meet-structure. Semilattices frequently appear when modeling partially ordered sets through algebraic operations.

4.2.2 Regular bands and connections

Some bands satisfy additional identities that reflect stronger compatibility among idempotents. Such subclasses are often studied because they connect to the geometry of ideal structures and to decompositions influenced by Green’s relations. Regularity within band-like environments provides a bridge between pure idempotent algebra and more general semigroup regularity.

4.3 Cancellative and Archimedean Conditions

4.3.1 Left/right cancellative semigroups

A semigroup is left cancellative if \(ax=ay\) implies \(x=y\), and right cancellative if \(xa=ya\) implies \(x=y\). Cancellation rules prevent certain kinds of collapsing behavior and can force stronger structural restrictions, particularly when combined with finiteness assumptions.

4.3.2 Natural order and comparability notions

In cancellative contexts, one can often define partial orders related to divisibility (e.g., \(a\) divides \(b\) if \(b=ac\) or \(b=ca\)). These relations help interpret algebraic data as order-theoretic information, clarifying which elements can reach others via multiplication.

4.4 Power Conditions and Stabilization Phenomena

4.4.1 Identities involving powers

Semigroup identities often involve powers of elements, such as requiring \(x^m=x^n\) for certain integers \(m>n\). Such constraints restrict how repeated multiplication behaves, limiting the kinds of cycles or growth patterns elements may have.

Power identities are especially useful for describing finite semigroups, where repeated products must eventually repeat.

4.4.2 Eventually periodic behavior in finite cases

In a finite semigroup, the sequence \(a,a^2,a^3,\dots\) cannot grow indefinitely without repetition. Consequently, powers of a given element become eventually periodic: after some point, multiplying by the element yields a repeating cycle of values. This stabilization phenomenon underlies many classification results for finite semigroups and contributes to algorithmic strategies for analysis.

5 Advanced Topics and Applications

5.1 Representation Theory (Semigroup Viewpoint)

5.1.1 Semigroup algebras

Given a semigroup \(S\) and a field \(K\), the semigroup algebra \(K[S]\) is formed by taking formal \(K\)-linear combinations of elements of \(S\), with multiplication extended from the semigroup operation bilinearly. Studying \(K[S]\)-modules translates semigroup questions into linear algebra and module theory.

5.1.2 Modules and actions

A representation in this context is often realized as a module over \(K[S]\), equivalently as an action of \(S\) by linear transformations. This approach reveals how semigroup elements act on vector spaces, enabling decompositions into invariant subspaces and the use of character-like tools where appropriate.

5.2 Automata-Theoretic Connections

5.2.1 Recognizable languages

In automata theory, a language can be recognized by a finite-state machine. The set of transformations induced by input words forms a finite semigroup called the transition semigroup. Language recognition therefore corresponds to semigroup behavior: different languages can be classified according to properties of the semigroup generated by their transitions.

5.2.2 Syntactic semigroups

The syntactic semigroup of a language is the quotient of the free semigroup by identifying words that cannot be distinguished by the language’s continuation behavior. It provides a minimal semigroup capturing the language’s combinational structure. This semigroup becomes a focal object for studying algebraic recognizability and for deriving logical or combinatorial characterizations.

5.3 Semigroup Varieties and Identities

5.3.1 Equational theories

An identity is an equation \(u=v\) between words that must hold under all substitutions of the variables by elements of a semigroup. The collection of all identities true in a semigroup forms its equational theory. Identities provide a language for defining and comparing semigroups through what algebraic patterns they obey.

5.3.2 Varieties and closure properties

A variety of semigroups is a class closed under forming homomorphic images, subsemigroups, and direct products. These closure properties mirror those in universal algebra: varieties represent semantic categories defined by identities. As a result, classification can often be approached by understanding which identities define which varieties.

5.3.3 Typical identities used to classify varieties

Classification typically relies on families of identities that reflect combinational features such as commutation-like behavior, idempotence patterns, or power stabilization. By selecting appropriate identities—sometimes involving repeated variables—one can distinguish varieties and build hierarchies of semigroup classes.

5.4 Complexity and Computation Notes

5.4.1 Computational questions about finite semigroups

Many interesting problems in semigroup theory restrict to finite structures, where computation is feasible. Examples include determining whether a given finite operation table forms a semigroup (associativity checking), enumerating ideals and Green’s classes, and testing regularity or cancellativity.

Finite settings also make it practical to explore identity satisfaction for proposed classes, and to investigate stabilization of powers.

5.4.2 Algorithms for congruences and ideals (high level)

Computing all congruences or all ideals can be challenging because the number of such structures can grow quickly with the size of the semigroup. Nevertheless, high-level algorithmic approaches often use closure operations, partition refinement ideas, and incremental construction based on multiplication constraints. In practice, special properties—such as regularity, finiteness, or restrictions given by identities—can significantly reduce search space and improve computational tractability.