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Semilattice sums of algebras and Mal'tsev products of varieties

This paper establishes that the Mal'tsev product of a strongly irregular variety $\mathcal{V}$ (lacking nullary operations but containing a non-unary operation) and the variety of semilattices $\mathcal{S}$ forms a variety consisting of semilattice sums of algebras from $\mathcal{V}$ , for which an equational base is derived, while noting that this property does not necessarily hold for regular varieties.

Original authors: Clifford Bergman, Tomasz Penza, Anna B. Romanowska

Published 2026-04-01

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CC BY 4.0

Original authors: Clifford Bergman, Tomasz Penza, Anna B. Romanowska

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master architect trying to build a new kind of city. You have two existing blueprints:

The "V" Blueprint: A strict, complex set of rules for building individual houses (these are your algebras). Maybe these houses have specific rules about how doors open or how rooms connect.
The "S" Blueprint: A simple, flexible map for arranging neighborhoods. This map is based on semilattices—think of it as a hierarchy where neighborhoods can merge, but they never split apart. It's like a family tree or a filing system where folders can contain other folders.

The paper asks a big question: If we build a city where every neighborhood follows the "S" map, and every single house inside those neighborhoods follows the "V" rules, does the whole city follow a single, consistent set of rules?

In mathematical terms, this is asking if the Mal'tsev product (the combination of these two structures) is a Variety (a perfectly consistent system governed by equations) or just a Quasivariety (a system that works mostly, but might need extra "if-then" rules to stay consistent).

The Main Discovery: The "Strongly Irregular" Key

The authors, Bergman, Penza, and Romanowska, discovered a special key that unlocks the door to a perfectly consistent city.

They found that if your "V" houses follow a specific, quirky rule called "Strongly Irregularity," the whole city works perfectly.

What is "Strongly Irregularity"?
Imagine a rule in your house blueprint that says: "No matter what you do, the left side of the equation always wins."
For example, a rule like Left(x, y) = x. It doesn't matter what y is; the result is always x.

Regular rules are like a fair negotiation: x + y = y + x. Both sides matter.
Strongly Irregular rules are like a dictator: x always dictates the outcome.

The Analogy of the "Dictator House":
If every house in your city has a "dictator" rule (Strongly Irregular), then when you stack these houses into neighborhoods (the Semilattice sum), the chaos doesn't spread. The "dictator" nature of the houses keeps the whole structure stable.

The paper proves that if your houses have this "dictator" rule and no "empty" rules (no nullary operations), then the entire city is a Variety. You can write down a single list of equations that describes the whole city perfectly.

The "Prolongation" Machine

How do you write the rules for this new city? The authors introduce a tool they call "Prolongation."

Think of Prolongation as a "Rule Amplifier."

You start with a rule for a single house: x + y = x.
The Prolongation machine takes that rule and expands it to cover the whole city. It says, "Okay, if this rule works for one house, it must work for a whole neighborhood, and a whole district, and the whole city."
It turns simple house rules into complex city-wide laws that account for the "neighborhood" structure.

The paper shows that if you take the "Dictator" houses and run them through this Prolongation machine, you get the exact set of rules needed to describe the entire city.

When Things Go Wrong: The "Regular" Trap

The paper also warns us what happens if the houses are Regular (fair and balanced, like x + y = y + x).

If you try to build a city of "fair" houses arranged in a "neighborhood" map, the system often breaks down. The rules for the whole city can't be written as a simple list of equations. You end up needing complex "if-then" conditions (Quasivarieties) to make it work.

The Counter-Example:
The authors show a specific case where you take the simplest "fair" system (Semilattices) and try to combine it with itself. The result is a messy structure that cannot be described by a simple list of equations. It's like trying to build a city where every rule depends on a specific exception; it becomes too complicated to be a "Variety."

The "Lallement Sum": Rebuilding the City

Finally, the paper discusses how to actually build these cities.

Plonka Sums: This is the "easy" way to build. You take your houses and glue them together using strict, predictable bridges. This works for "Regular" systems.
Lallement Sums: This is the "advanced" way. It's like building a city where the bridges between neighborhoods are flexible. You can stretch and adapt the connections based on the specific needs of the houses.

The authors show that for "Dictator" houses (Strongly Irregular), you can use a special version of the Lallement Sum (called a "Strict" Lallement Sum) to perfectly reconstruct the city from its parts. It's like having a blueprint that tells you exactly how to assemble the city block by block, ensuring the "dictator" rules hold up everywhere.

Summary in a Nutshell

The Problem: Can we combine complex rules (Varieties) with simple hierarchical structures (Semilattices) to make a new, consistent system?
The Solution: Yes, IF the complex rules have a "dictator" nature (Strongly Irregularity).
The Method: Use a "Rule Amplifier" (Prolongation) to turn house rules into city rules.
The Warning: If the rules are "fair" (Regular) instead of "dictatorial," the system might break and become too messy to be a simple Variety.
The Result: We now have a clear recipe for building these complex mathematical cities, provided we use the right kind of "dictator" bricks.

This paper is essentially a guide for mathematicians on how to safely mix complex algebraic structures with hierarchical ones, ensuring the result is stable and predictable.

1. Problem Statement

The paper addresses a fundamental question in universal algebra regarding the Mal'tsev product of two varieties of algebras, denoted as $Q \circ R$ .

Context: It is a known result (Mal'tsev, 1970s) that the Mal'tsev product of two varieties is always a quasivariety (a class defined by quasi-identities), but not necessarily a variety (a class defined by identities).
Specific Focus: The authors investigate the specific case where $R$ is the variety of semilattices ( $S$ ) and $Q$ is an arbitrary variety $V$ of the same similarity type. The resulting class, $V \circ S$ , consists of semilattice sums of $V$ -algebras (algebras that are disjoint unions of $V$ -subalgebras over a semilattice quotient).
The Core Question: Under what conditions on the variety $V$ is the Mal'tsev product $V \circ S$ guaranteed to be a variety (i.e., axiomatizable by equations alone) rather than just a quasivariety?

2. Methodology and Key Concepts

The authors employ a combination of structural decomposition, equational logic, and the theory of free algebras.

Strongly Irregular Varieties: A variety $V$ is called strongly irregular if it satisfies an identity of the form $t(x, y) = x$ for some binary term $t$ . This contrasts with regular varieties, where every identity has the same variables on both sides.
Prolongation of a Variety ( $V^p$ ): The authors introduce a construction called the prolongation. Given a variety $V$ $V$ with an equational base $\Sigma$ $Σ$ , the prolongation $V^p$ $V^{p}$ is the variety defined by the set of identities $\Sigma^p$ $Σ^{p}$ .
- Mechanism: For any identity $\sigma \in \Sigma$ , $\sigma^p$ is generated by replacing variables in $\sigma$ with terms $r_i$ that contain all variables of a specific arity. This process transforms potentially irregular identities into regular ones that hold in the semilattice sum structure.
Semilattice Sums and Congruences: The analysis relies heavily on the semilattice replica congruence ( $\varrho$ ), the smallest congruence such that the quotient $A/\varrho$ is a semilattice. In strongly irregular varieties, this congruence is unique and ensures that each congruence block is a subalgebra in $V$ .
Płonka Sums vs. Semilattice Sums: The paper distinguishes between:
- Płonka Sums: A specific type of semilattice sum where the connecting homomorphisms form a functor (covariant functor from the semilattice to the category of algebras). These form the regularization ( $\tilde{V}$ ) of a variety.
- Semilattice Sums (General): A broader class where the connecting maps are not necessarily functorial. These form the Mal'tsev product $V \circ S$ .
- Pseudo-regularization ( $\hat{V}$ ): An intermediate variety defined by specific identities (P1–P4) that captures semilattice sums without requiring full functoriality.

3. Key Contributions and Results

A. The Main Theorem (Theorem 5.3)

The central result of the paper establishes a sufficient condition for the Mal'tsev product to be a variety:

Theorem: If $V$ is a strongly irregular variety (satisfying $t(x,y)=x$ ) with no nullary operations and at least one non-unary operation, then the Mal'tsev product $V \circ S$ is a variety.

Furthermore, $V \circ S$ coincides exactly with the prolongation $V^p$ . The equational base for $V \circ S$ is derived directly from the equational base of $V$ via the prolongation construction ( $\Sigma^p$ ).

B. Structural Decomposition of Free Algebras

The authors prove that the free algebra of the prolonged variety $V^p$ is itself a semilattice sum of $V$ -algebras.

Proposition 3.2: The free $V^p$ -algebra over a set $X$ is isomorphic to a semilattice sum $\bigoplus_{s \in S} A_s$ , where $S$ is the free semilattice over $X$ , and each component $A_s$ is a free $V$ -algebra.
This structural insight allows the authors to prove that any homomorphic image of a free $V^p$ -algebra (which defines the variety $V^p$ ) remains within the Mal'tsev product $V \circ S$ .

C. Necessity of Strong Irregularity

The paper demonstrates that the condition of strong irregularity is essential.

Counterexamples: If $V$ is a regular variety (e.g., commutative groupoids, semilattices themselves), $V \circ S$ is generally not a variety.
Example 4.13: The authors construct a specific algebra in $S \circ S$ (the product of semilattices with themselves) that fails a specific quasi-identity, proving that $S \circ S$ is not a variety. This implies that for regular varieties, the Mal'tsev product cannot be axiomatized purely by equations.

D. Representation via Lallement Sums

The paper extends the representation theory of these algebras:

While Płonka sums represent the regularization (where connecting maps are functorial), the authors utilize Lallement sums to represent the more general semilattice sums found in $V \circ S$ .
Theorem 7.2: For varieties with units (or specific entropic properties), semilattice sums can be reconstructed as strict Lallement sums. This provides a concrete method to build algebras in $V \circ S$ from their components and the semilattice quotient.

4. Applications and Examples

The authors apply their theory to several specific algebraic structures:

**Left-Zero Bands ($LZ $):**$ LZ \circ S$ is a variety (specifically $LZ^p$ ), but its members are not necessarily semigroups (associative).
Steiner Quasigroups ($SQ$): The Mal'tsev product $SQ \circ S$ is a variety ( $SQ^p$ ). The authors distinguish between the regularization (Płonka sums) and the pseudo-regularization (general semilattice sums), showing they are distinct.
Birkhoff Systems and Lattices: The class of Birkhoff systems that are semilattice sums of lattices is identified as $L \circ S$ .
Groups ($GP$): The Mal'tsev product of groups with semilattices is a variety of groupoids (not semigroups), distinct from the class of Clifford semigroups (which are Płonka sums of groups).
Barycentric Algebras: The paper discusses how barycentric algebras are semilattice sums of convex sets, illustrating the distinction between the variety and its quasivariety closure.

5. Significance

Resolution of a Long-standing Question: The paper provides a definitive answer to when the Mal'tsev product with semilattices yields a variety, resolving a gap in the literature where it was previously unknown for general varieties.
New Axiomatization Technique: The introduction of the prolongation ( $V^p$ ) offers a systematic method to generate equational bases for complex Mal'tsev products from the bases of the constituent varieties.
Refinement of Decomposition Theory: By distinguishing between Płonka sums (regularization) and general semilattice sums (Mal'tsev product), the paper clarifies the structural hierarchy of algebraic varieties. It shows that while Płonka sums are a subset of semilattice sums, the latter form a variety only under strong irregularity.
Bridge to Representation Theory: The connection to Lallement sums provides a constructive framework for understanding the internal structure of these algebras, moving beyond abstract existence proofs to concrete representations.

In summary, the paper establishes that strong irregularity is the key property that ensures the Mal'tsev product of a variety with semilattices remains a variety, and it provides the tools (prolongation and Lallement sums) to describe and construct these algebras explicitly.