1 Definitions and basic intuition

1.1 Regular elements in semigroups

A semigroup is a set equipped with an associative multiplication. In this setting, “regularity” captures the idea that an element is not intrinsically blocked by the multiplication: it can be funneled back to itself after multiplying by a suitable “witness” element.

1.1.1 Inner inverse formulation

An element \(a\) of a semigroup \(S\) is (von Neumann) regular if there exists an element \(b\in S\) such that \[ a = aba. \] The element \(b\) is often called an inner inverse of \(a\) (with respect to the semigroup multiplication). The equation states that composing with \(b\) and then again with \(a\) recovers \(a\).

1.1.2 Alternative equivalent characterizations

Several equivalent formulations are commonly used. For instance, regularity can be expressed through the existence of matching factorizations:

  • The principal right ideal \(aS\) and the principal left ideal \(Sa\) interact in a way that allows \(a\) to lie in their product.
  • In finite semigroups, regularity can be linked to properties of the directed graph underlying the multiplication (equivalently, to reachability patterns produced by repeated multiplication).

In practice, one typically proves equivalence by starting from \(a=aba\) and manipulating to obtain the desired ideal or factorization conditions, and conversely reconstructing an element \(b\).

1.2 Regular elements in rings and algebras

In rings, the same basic idea appears but the algebraic environment changes what “inverse-like” means because addition and additive structure become available. The definition also shifts slightly depending on which regularity notion is used.

1.2.1 Von Neumann regular elements

Let \(R\) be a ring (not necessarily commutative). An element \(a\in R\) is von Neumann regular if there exists \(b\in R\) with \[ a = aba. \] Thus, the same equation defines regularity in rings as it does in semigroups when only multiplication is considered. However, ring structure makes additional interpretations natural, particularly via ideals and module homomorphisms.

1.2.2 Generalized inverse perspective

The ring viewpoint often frames \(b\) as a form of generalized inverse: while \(b\) typically will not satisfy \(ab=ba=1\), it compensates for the failure of invertibility by ensuring that \(a\) acts like a “retract” of itself within the ring’s multiplicative behavior. This perspective is especially useful in operator algebra and in endomorphism rings, where multiplication corresponds to composition.

1.3 Variants and naming conventions

The term “regular element” is widely used, but different authors may emphasize different regularity conditions depending on context.

1.3.1 (von Neumann) regular vs. other regularity notions

The phrase (von Neumann) regular is used to distinguish the \(a=aba\) condition from other regularity concepts. For example, one may encounter “regularity” in settings such as:

  • strong regularity, where an additional constraint on the witness element is required;
  • regularity notions tied to involutions or *-operations, where the generalized inverse interacts with a conjugate-linear structure;
  • regularity in non-associative algebraic systems, where the product is not necessarily associative.

Because these notions share a family resemblance, clear naming is important when importing results between subfields.

1.3.2 Regularity in *-algebras (outline level)

In a *-algebra (a complex or real algebra with an involution \(x\mapsto x^*\)), one can refine the notion of regularity by requiring compatibility of the generalized inverse with the involution. At a high level, this leads to “inverse-like” conditions such as the existence of a \(b\) built from \(a\) and \(a^*\) that satisfies a regularity equation together with an adjoint-related constraint. The exact formulation depends on the convention (e.g., Moore–Penrose-type conditions versus weaker involution-compatible regularities).

2 Structural properties

2.1 Idempotents and regular elements

Idempotent elements play a central role in regularity theory because they behave like stable “projections” under multiplication.

2.1.1 Relationship to idempotent elements

If \(e\) is idempotent (\(e^2=e\)), then \(e\) is automatically regular: taking \(b=e\) yields \(e=ebe\). Conversely, many regularity statements are proved by constructing idempotents related to a given regular element. Intuitively, the witness equation \(a=aba\) often forces \(ab\) or \(ba\) to act like idempotents or to generate idempotent behavior.

2.1.2 Regularity via idempotent-generated behavior

In many semigroup and ring contexts, regularity of \(a\) is closely tied to the presence of idempotents inside principal ideals generated by \(a\). For instance, if \(a\) is regular, then the right ideal \(aS\) typically contains an idempotent, and similarly for the left ideal. These idempotents can be constructed explicitly from \(a\) and its witness \(b\) (often using \(ab\) or \(ba\) and checking the defining relations).

2.2 Green’s relations (semigroup viewpoint)

Green’s relations organize a semigroup by comparing generated ideals. They provide a natural language for describing how regularity behaves within the semigroup’s internal structure.

Green’s relations include \(\mathcal{L}\), \(\mathcal{R}\), and their join \(\mathcal{D}\). Regular elements interact strongly with these classes: within a \(\mathcal{D}\)-class, regularity can often be detected by checking the presence of idempotents. Because \(\mathcal{D}\)-classes reflect two-sided ideal structure, this makes regularity a property that can be understood “up to the appropriate ideal level.”

2.2.2 Consequences for principal ideals

For a regular element \(a\), the principal ideals it generates (left, right, and two-sided) tend to have more internal symmetry than arbitrary principal ideals. For example:

  • the \(\mathcal{R}\)-class of a regular element frequently contains idempotents;
  • the \(\mathcal{L}\)-class likewise contains idempotents;
  • the structure of \(aS\) and \(Sa\) becomes more tractable for computations and classifications.

These consequences are important in finite semigroup theory and in algorithmic analyses of regular components.

2.3 Regularity under homomorphisms

Regularity is not simply an absolute property of an element independent of context; it interacts with morphisms between algebraic structures.

2.3.1 Images of regular elements

If \(\varphi:S\to T\) is a semigroup homomorphism and \(a\) is regular in \(S\), then \(\varphi(a)\) is regular in \(\varphi(S)\) (and hence in \(T\) if the image is viewed as a subsemigroup). The witness carries over: if \(a=aba\), then applying \(\varphi\) gives \(\varphi(a)=\varphi(a)\varphi(b)\varphi(a)\).

An analogous statement holds for rings: ring homomorphisms preserve the multiplicative equation \(a=aba\).

2.3.2 Preimages and saturation properties

Going backwards is subtler. A preimage of a regular element need not be regular in general, because the necessary witness might fail to exist in the domain even if one exists after mapping. However, under additional hypotheses—such as surjectivity, compatibility with decompositions, or restrictions to certain substructures—preimage regularity can be recovered in a controlled way. In categorical terms, regularity is often “stable under the direction of morphisms” where witness equations can be transported.

3 Examples

3.1 Semigroup examples

3.1.1 Transformation semigroups

A transformation semigroup consists of functions on a set with composition as multiplication. Let \(a\) be a function. Regularity means there exists a function \(b\) such that \[ a = a\circ b \circ a. \] A useful intuition is that \(b\) can “choose” representatives in the image of \(a\) so that applying \(a\) again reproduces the original effect of \(a\). In many transformation semigroup settings, regularity can be characterized using properties of images and fibers: roughly, the map must be compatible with a section-like behavior on the image.

3.1.2 Matrix semigroups and partial maps

Matrix semigroups under multiplication supply concrete algebraic examples. For a square matrix \(A\), regularity in the semigroup sense asks for a matrix \(B\) with \(A=ABA\). Over a field, such equations relate to rank and to the existence of suitable decompositions of the vector space into kernel and complement parts. Closely related reasoning applies to semigroups of partial functions (where the multiplication corresponds to composition with domain restrictions): regularity corresponds to the ability to “factor” the map so that composing twice matches the original.

3.2 Ring and algebra examples

3.2.1 Endomorphism rings of modules

Let \(M\) be a module and consider \(R=\mathrm{End}(M)\). An endomorphism \(a\in \mathrm{End}(M)\) is regular when there exists another endomorphism \(b\) with \(a=aba\). This equation can be interpreted in terms of module homomorphisms: it often implies that \(M\) decomposes (at least partially) into submodules linked to the image and kernel of \(a\). Regularity then becomes a statement about how the image of \(a\) behaves under the action induced by \(a\) and \(b\).

3.2.2 Full matrix rings

For \(R=M_n(F)\), von Neumann regularity is governed by linear algebraic structure. Over a field \(F\), every matrix is von Neumann regular, because one can construct \(B\) using a rank factorization and appropriate choices of complements so that \(A=ABA\) holds. This example is a benchmark: it highlights that regularity is sometimes automatic in “well-behaved” algebraic environments.

3.3 Non-examples and boundary cases

3.3.1 When regularity fails

In rings with more complicated ideal structures, regularity can fail. An element \(a\) can fail to satisfy \(a=aba\) because no element \(b\) exists to correct the mismatch between the action of \(a\) and the multiplication structure of the ring. In practice, failures are often detected by ideal-theoretic obstructions: if the principal ideal generated by \(a\) cannot be expressed in the regular manner predicted by \(a=aba\), then regularity does not hold.

3.3.2 Dependence on the chosen ambient structure

An element might be regular in one ambient algebra but not in a larger one, because the witness \(b\) may lie in the smaller structure but not in the larger. Conversely, enlarging the algebra can introduce additional elements that provide a witness. Therefore, regularity is relative to the ambient algebraic structure and its available elements, even if the same underlying multiplication-like operation is extended.

4 Regularity criteria and tests

4.1 Constructing the witness element

4.1.1 Finding an inner inverse (semigroups)

In a semigroup, finding \(b\) such that \(a=aba\) is the operational content of regularity. In concrete classes like transformation semigroups or finite semigroups, one can often build \(b\) by specifying its action on the image of \(a\) in a way that ensures the recovery equation. In abstract settings, one instead uses ideal relations: if the necessary idempotent exists in a principal ideal generated by \(a\), it can be used to derive a valid witness.

4.1.2 Finding a generalized inverse (rings)

For rings, once an element \(b\) exists with \(a=aba\), it serves as a generalized inverse. In modules and endomorphism rings, one can attempt to define \(b\) using decompositions of \(M\) tied to \(a\)’s image and kernel. In matrix rings over fields, \(b\) can be constructed via rank factorizations and explicit linear-algebra formulas.

4.2 Equivalent conditions in common classes

4.2.1 Conditions using ideals/submodules

In rings, von Neumann regularity can be characterized by ideal properties. For example, a standard viewpoint is that the principal right ideal \(aR\) and principal left ideal \(Ra\) behave compatibly, often with the consequence that certain quotient structures become semisimple-like or that the principal ideal generated by \(a\) contains an idempotent element in a controlled way. In endomorphism rings, these ideal conditions translate into statements about the submodules \(\mathrm{Im}(a)\) and \(\ker(a)\).

4.2.2 Conditions using factorization patterns

Another common approach is to repackage \(a=aba\) as a factorization statement. If \(a\) is regular, it can be written in a way that reflects a “sandwich” pattern: \(a\) factors through a component that permits a retraction after multiplication by the witness. This style of criterion is especially useful for semigroups of functions and for matrix semigroups, where factorization corresponds to splitting along images or ranks.

4.3 Computation in finite settings

4.3.1 Algorithmic search for inner/generalized inverses

In finite semigroups or finite rings, regularity can be decided by brute force or by pruning. One checks whether there exists \(b\) with \(a=aba\). While complexity grows quickly with size, the method is straightforward and forms the basis for experimental investigation and for verifying conjectures in small structures.

4.3.2 Using canonical forms in matrices

For finite-dimensional matrix algebras over fields, one can use canonical decompositions (such as rank or similarity classifications, depending on the exact setting). A regularity witness can then be computed systematically from the decomposition rather than searched blindly. This approach is faster and more informative because it explains why regularity holds (e.g., via the ability to construct complements to kernels).

5 Algebraic consequences

5.1 Regularity and cancellation-like behavior

Regular elements often exhibit behavior reminiscent of cancellation, though not in the literal sense that semigroup or ring cancellation laws would provide.

5.1.1 Impact on divisibility relations

In semigroups, a regular element tends to have more stable divisibility relationships through Green’s relations. While cancellation may fail globally, regularity can ensure that certain “one-sided” divisibility patterns stabilize: products involving a regular element can often be related back to the element through the witness equation, constraining how ideals compare and preventing some degeneracies.

5.1.2 Stability under certain restrictions

Regularity is typically preserved when restricting to subsemigroups or passing to suitable substructures where the witness element remains available. Likewise, in ring settings, regular elements behave predictably when acting on modules where the endomorphism ring inherits the relevant decompositions.

5.2 Regular elements and decomposition ideas

The identity \(a=aba\) can be interpreted as saying that multiplication by \(a\) is compatible with a retraction mediated by \(b\). In module and endomorphism settings, this frequently yields decompositions of the module into parts controlled by \(\ker(a)\), \(\mathrm{Im}(a)\), and associated complement submodules. Such splittings are central to why regularity is useful: it converts an abstract equation into geometric structure.

5.2.2 Regular components in structured objects

In semigroup theory, regular elements often organize the semigroup into layers: regular \(\mathcal{D}\)-classes contain idempotents and support stronger structural theorems. In ring theory, regular elements can also influence how the ring decomposes into parts governed by regularity, especially in classes of rings where every element is regular.

5.3 Connections to module theory

5.3.1 Regular elements in endomorphism rings

When \(a\) is a module endomorphism, the equation \(a=aba\) for some endomorphism \(b\) translates into a statement about how \(a\) behaves relative to invariant submodules. The witness \(b\) can be seen as implementing an internal correction that replays the effect of \(a\) after passing through a suitable intermediate stage.

5.3.2 Interpretations via kernels and images

In many practical cases, especially over fields or in well-structured module categories, regularity can be analyzed through kernels and images. For instance, constructing \(b\) often amounts to choosing a complement to the kernel and then defining \(b\) as a pseudo-inverse on that complement, extended suitably. This viewpoint connects regularity to classical “inverse on the range” ideas, even when the ring lacks a genuine inverse.

6 Advanced generalizations

6.1.1 Strengthening the defining condition

Strongly regular elements impose an additional requirement beyond \(a=aba\). While the precise strengthening varies by definition, the theme is that the witness element \(b\) must interact with \(a\) in a more rigid way, such as satisfying extra commuting or idempotent-related constraints. This strengthens the structural consequences and can simplify classification in certain algebraic settings.

6.1.2 Impact on classification

Stronger regularity narrows the class of elements and can lead to finer invariants for semigroups and rings. As a result, classification theorems become sharper: one can distinguish objects that are regular but not strongly regular, and thereby obtain more refined decompositions.

6.2 Regularity in categorical/algebraic frameworks

6.2.1 Algebra objects with generalized inverses

In categorical treatments, one can view generalized inverses as morphisms that satisfy equations analogous to \(a=aba\), but interpreted within a category where “multiplication” becomes composition. This allows regularity to be studied in settings beyond rings and semigroups, provided an appropriate composition-like operation and notion of morphism are available.

6.2.2 Connections to inverse-like morphisms

These categorical generalizations emphasize that regularity is about the existence of morphisms acting as retractions or splitting maps. As a consequence, many proofs can be organized using universal properties, pullbacks, or splitting lemmas rather than direct algebraic manipulation.

6.3 Regular elements in non-associative contexts (scope note)

6.3.1 How definitions may change

If associativity fails, the expression \(a=aba\) may not even be well-defined without choosing a parenthesization, and different parenthesized forms may not be equivalent. Therefore, regularity notions in non-associative algebras require adjusted definitions tailored to the algebra’s multiplication laws.

6.3.2 Typical obstacles and adjustments

Common obstacles include:

  • the loss of straightforward manipulations used in associative proofs;
  • difficulties in translating ideal-based criteria;
  • the need for additional axioms to ensure that “witness” elements behave consistently with the algebra’s identities.

As a result, regularity theory in non-associative settings is more delicate and often develops new techniques to compensate for the absence of associativity.

7 Exercises and quick check problems

7.1 Prove regularity using explicit witnesses

Given a concrete semigroup (e.g., transformations on a finite set) or ring (e.g., a matrix ring), find \(b\) for a specified element \(a\) and verify the equation \(a=aba\) directly.

7.2 Show non-regularity in small examples

Pick a small semigroup or ring and demonstrate that for a candidate element \(a\), no element \(b\) satisfies \(a=aba\). In finite cases, the task can be completed by exhaustive checking.

7.3 Verify invariance under isomorphisms and homomorphisms

Let \(f:S\to T\) be a homomorphism. Prove that if \(a\) is regular in \(S\), then \(f(a)\) is regular in \(T\). Then test whether the converse can fail by constructing examples.

7.4 Compute regular elements in finite semigroups/rings

For a given finite semigroup or finite ring presented by its multiplication table, determine the subset of regular elements. Record how the set of regular elements relates to idempotents and Green’s relations (in semigroups).