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Two nonfinitely based additively idempotent semirings of order four

This paper establishes sufficient conditions for nonfinite basis in additively idempotent semirings, proving that two specific four-element semirings lack finite bases, their generated variety interval contains continuum many distinct varieties, and the join of finitely based varieties is not necessarily finitely based, while also identifying the smallest example of a finitely based semiring whose zero-extension is nonfinitely based.

Original authors: Mengya Yue, Miaomiao Ren, Zidong Gao

Published 2026-03-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mengya Yue, Miaomiao Ren, Zidong Gao

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 specific type of structure called a Semiring. In the world of mathematics, a semiring is like a rulebook for a game involving two actions: Adding things together and Multiplying them.

In this specific paper, the authors are studying a special version of this game called an AI-Semiring (Additively Idempotent). The "Additively Idempotent" rule is a bit like a magic spell: if you add a number to itself, it doesn't get bigger; it stays the same ( $A + A = A$ ). Think of it like a bucket of water: if you pour a cup of water into a bucket that's already full, the level doesn't rise; it just stays full.

The Big Mystery: The "Finite Basis" Problem

The central question the authors are asking is: "How many rules do we need to write down to describe every possible structure in this game?"

Finitely Based: Imagine you can write down a short, finite list of rules (like a cheat sheet) that covers every possible move in the game. If you have this list, you know everything about the game.
Non-Finitely Based: Imagine the game is so complex that no matter how long your list of rules is, there will always be a new, weird move that your list doesn't cover. You would need an infinite list of rules to describe it perfectly.

For decades, mathematicians have been trying to figure out which small, simple games (semirings with very few elements) have a finite list of rules and which ones require an infinite one.

The Discovery: Two Tiny Giants

The authors focused on two specific, tiny games, each with only 4 elements (like a 4-sided die). They named them S(4,545) and S(4,634).

You might think, "Hey, they only have 4 elements! That's tiny. They must have a simple, short list of rules."

The authors proved the opposite. They discovered that despite being tiny, these two games are actually monsters of complexity.

They are Non-Finitely Based.
To describe them, you need an infinite list of rules. You can never finish writing the rulebook.

They developed two new "detective tools" (mathematical conditions) to prove this. It's like finding a new way to look at a fingerprint and realizing, "Oh, this tiny print actually belongs to a criminal with an infinite record."

The "Join" Surprise: Mixing Simple Things Makes a Monster

Here is the most surprising part of the paper, which answers a big question in the math world.

Imagine you have two simple, well-behaved games (let's call them Game A and Game B). Both of them have short, finite rulebooks.

Game A is simple.
Game B is simple.

The big question was: If you mix Game A and Game B together to create a new, bigger game (a "Join"), will the new game still be simple?

Most people guessed "Yes." They thought mixing two simple things should result in something simple.

The authors proved: NO.

They showed that if you mix two specific simple games, the result is a monster (a non-finitely based game).

They took a simple 3-element game (S53) and a simple 2-element game (D2).
They mixed them.
The result was S(4,545), which is a monster with an infinite rulebook.

Analogy: Imagine you have a box of simple Lego bricks (Game A) and a box of simple wooden blocks (Game B). You can describe both boxes with a short manual. But when you combine them into one giant box, the instructions for building everything in that giant box become infinite. You can't write a manual that fits on a single page anymore.

The "Extension" Puzzle

The paper also looked at what happens when you add a "zero" element to a game.

They took a simple 3-element game (S53) that had a finite rulebook.
They added a new "zero" element to it, creating a 4-element game (which turned out to be S(4,634)).
Result: The new, slightly bigger game became a monster with an infinite rulebook.

This is the smallest example ever found where adding just one tiny piece to a simple system breaks the simplicity entirely.

The "Lattice" of Varieties

Finally, the authors looked at the space between these two games. In math, there's a concept called a "lattice" (like a grid) that organizes all these games based on how complex they are.

They looked at the space between the "Join" game (S(4,545)) and the "Extension" game (S(4,634)).

They proved that inside this tiny gap, there are uncountably infinite different types of games.
Analogy: Imagine a hallway between two doors. You might expect to find a few rooms in between. But the authors proved that this hallway actually contains an infinite number of distinct, unique rooms, each with its own set of rules.

Why Does This Matter?

It breaks intuition: It shows that complexity doesn't always come from size. A tiny system can be infinitely complex.
It solves a puzzle: It answers a long-standing question about whether mixing simple systems always keeps them simple (the answer is no).
It provides tools: The authors created new "detective tools" that other mathematicians can use to find more of these hidden monsters in the world of algebra.

In short: The authors found two tiny, 4-element math games that are secretly infinitely complex. They proved that mixing two simple games can create a monster, and that adding a single zero to a simple game can turn it into a monster. It's a reminder that in mathematics, the smallest things can sometimes hold the biggest secrets.

1. Problem Statement and Context

The paper addresses the Finite Basis Problem (FBP) within the theory of additively idempotent semirings (ai-semirings). An ai-semiring is an algebra $(S, +, \cdot)$ where $(S, +)$ is a commutative idempotent semigroup, $(S, \cdot)$ is a semigroup, and the distributive laws hold. The FBP asks whether the equational theory (identities) of a given finite algebra can be derived from a finite set of identities.

Specific Goals:

To resolve the finite basis status of two specific 4-element ai-semirings, $S(4,545)$ and $S(4,634)$ , whose additive reducts are 4-element chains.
To investigate Problem 1.2: Is the join of two finitely based ai-semiring varieties necessarily finitely based?
To investigate Problem 1.3/1.4: Determine the cardinality of the interval $[V(S), V(S^0)]$ in the lattice of semiring varieties, where $S^0$ is the extension of $S$ by a zero element, and specifically whether the finite basis property is preserved under this extension.

2. Methodology

The authors employ a combination of syntactic methods and equational logic tools:

Term Construction: They define specific families of terms, denoted $u_{n,k}$ and $u_{n,n}$ , involving variables and specific structural constraints (e.g., linearity, content intersections).
Freeness Conditions: They utilize the concept of a term being " $t$ -free" (meaning $t$ is not a subterm of the term under any substitution).
Compactness Theorem: They apply the compactness theorem of equational logic (Lemma 2.3). The core strategy is to construct an infinite equational basis for the varieties generated by the target semirings and prove that no finite subset of this basis can define the variety. This is achieved by showing that for any finite subset, there exists a specific identity in the infinite set that is not derivable from the finite subset due to "freeness" properties.
Inequality Characterization: They characterize identities in terms of inequalities ( $u \preceq v$ meaning $u+v \approx v$ ) and analyze the content ( $c(u)$ ), length ( $\ell(u)$ ), and subword structures ( $S_2(u)$ ) of terms to determine derivability.
Variety Join Analysis: They analyze the relationship between the varieties generated by $S(4,545)$ , $S(4,634)$ , the 3-element semiring $S_{53}$ , and the 2-element distributive lattice $D_2$ .

3. Key Contributions and Results

A. Two New Sufficient Conditions for Non-Finite Basis

The authors establish two general sufficient conditions (Theorem 3.3 and Theorem 4.2) for an ai-semiring to be nonfinitely based. These conditions rely on the existence of an infinite set of inequalities $\Omega$ such that:

The variety is defined by a basis containing an infinite subset of $\Omega$ .
For any term $t$ in the basis (outside $\Omega$ ), a specific "hard" term $u_{m,k}$ is $t$ -free.
This structure prevents any finite subset of the basis from deriving the infinite set of necessary identities.

B. Resolution for $S(4,545)$ and $S(4,634)$

$S(4,545)$ : The authors prove that $V(S(4,545))$ $V (S (4, 545))$ is nonfinitely based.
- They show $V(S(4,545)) = V(S_{53}, D_2)$ .
- They provide an explicit infinite equational basis involving identities like $x^3 \approx x^2$ , $x \preceq x^2$ , $xy \preceq x + y^2z$ , and an infinite family of inequalities $\sigma_{n,k}$ .
- They prove that $S(4,545)$ contains $S_7$ (a known nonfinitely based 3-element semiring) and $S_{53}$ .
$S(4,634)$ : The authors prove that $V(S(4,634))$ $V (S (4, 634))$ is nonfinitely based.
- They identify $S(4,634)$ as isomorphic to $S_{53}^0$ (the extension of $S_{53}$ by a zero element).
- They provide a distinct infinite equational basis involving terms from the set $\Theta_n$ .

C. The Join of Finitely Based Varieties

The paper provides a negative solution to Problem 1.2:

It is proven that $V(S(4,545))$ is the join of two finitely based varieties: $V(S_{53})$ and $V(D_2)$ .
Since $V(S(4,545))$ is nonfinitely based, this demonstrates that the join of two finitely based ai-semiring varieties is not necessarily finitely based.

D. Cardinality of Variety Intervals

The authors determine the cardinality of the interval $[V(S(4,545)), V(S(4,634))]$ :

They prove this interval contains $2^{\aleph_0}$ (continuum many) distinct varieties.
This result extends to the interval $[V(S_{53}), V(S_{53}^0)]$ .
Significance: In previous examples of such intervals, both endpoints were nonfinitely based. Here, the left endpoint $V(S_{53})$ is finitely based, while the right endpoint $V(S_{53}^0)$ (isomorphic to $V(S(4,634))$ ) is nonfinitely based.

E. Smallest Counterexample for Extension

The paper identifies $S_{53}$ (a 3-element finitely based ai-semiring) as the smallest example (by order) of an ai-semiring $S$ such that $S$ is finitely based, but its zero-extension $S^0$ is nonfinitely based.
This resolves a specific case of Problem 1.3 regarding the preservation of the finite basis property under the $S \to S^0$ extension.

4. Significance

Classification Completion: This work significantly advances the classification of 4-element ai-semirings. While the finite basis status of 3-element semirings was largely resolved (with $S_7$ being the unique nonfinitely based exception), this paper tackles the most complex class of 4-element semirings (those with chain additive reducts).
Theoretical Breakthrough: The construction of a nonfinitely based variety as the join of two finitely based ones is a major theoretical result in universal algebra, challenging the intuition that "nice" properties (like finite basis) are preserved under joins.
Methodological Advancement: The syntactic conditions developed for proving non-finite basis are robust and applicable to other classes of algebras, offering new tools for researchers in the field.
Structural Insight: The results clarify the relationship between the finite basis property and the structural operations of adjoining a zero element or taking joins, highlighting that these operations can drastically increase the complexity of the equational theory.

In summary, the paper successfully proves that two specific 4-element ai-semirings are nonfinitely based, uses this to disprove the closure of finitely based varieties under joins, and establishes the existence of a continuum of varieties between a finitely based algebra and its nonfinitely based zero-extension.