Tic-Tac-Toe Matroid

• Tic-Tac-Toe matroid is a sparse-paving matroid defined on a 3x3 grid, exhibiting unique non-representability and duality phenomena.
• It features intricate circuit-hyperplane structures and extension properties that challenge linear, folded-linear, and algebraic representations.
• Its study deepens understanding of pseudomodularity and duality, providing new excluded-minor candidates in modern matroid theory.

The Tic-Tac-Toe matroid, often abbreviated as the TTT-matroid, occupies a central role in the paper of sparse-paving matroids, non-representability phenomena, and the interplay between matroid duality and algebraic representability. It is a paradigmatic example for several frontier concepts, including pseudomodularity, extension properties, and newly-constructed infinite matroid families that escape established classes such as linear and algebraic matroids.

1. Definition and Combinatorial Structure

The TTT-matroid arises as a sparse-paving matroid of rank 5 on a 9-element ground set, canonically labeled either as the $3 \times 3$ grid $E = \{a_1,a_2,a_3, b_1,b_2,b_3, c_1,c_2,c_3\}$ or equivalently as the points of the affine plane $F_3 \times F_3$ over the 3-element field, indexed $(x,y)$ with $x,y \in F_3 = \{-1,0,1\}$, $-1 \equiv 2 \bmod 3$ (Hochstättler, 13 Nov 2025, Bamiloshin et al., 2023).

A key construction proceeds via a rank-4 paving matroid $M_3$ on these nine points, defined by its 4-element "circuit-hyperplanes": $$\{a_i,a_j,b_i,b_j\},\quad \{a_i,a_j,c_i,c_j\},\quad \{b_i,b_j,c_i,c_j\},\quad 1 \le i < j \le 3$$ with the exception that $\{a_1,a_3,c_1,c_3\}$ is omitted, becoming independent. The TTT-matroid $T$ is the dual $T = M_3^*$, a rank-5 paving matroid whose circuit-hyperplanes are the 5-point complements of the above circuits. Dually in the geometric setting, the circuit-hyperplanes of $T$ correspond to the eight "row–column" unions $A_i \cup B_j$ where $A_i = \{(x,y) : y = i\}$ (rows) and $B_j = \{(x,y): x = j\}$ (columns), omitting $(i,j) = (0,0)$, or, stated differently, relaxing one circuit-hyperplane from the uniform configuration $T_1^3$ (Bamiloshin et al., 2023).

2. Circuit Structure and Family Enumeration

The family of TTT-matroids is characterized as all sparse-paving rank-5 matroids on nine points with circuit-hyperplanes containing the eight $A_i \cup B_j$ not including $(0,0)$. There exist precisely 181 non-isomorphic such matroids, forming two maximal classes:

• Type I: Matroids with all $A_i \cup B_j$ and all "diagonal" $C_k \cup D_\ell$ ($C_k = \{x-y = k\}$, $D_\ell = \{x+y = \ell\}$), for a total of 18 circuit-hyperplanes.
• Type II: Matroids containing the eight $A_i \cup B_j$, an extra 5-set of the form $(A_0 \cup B_0 \setminus \{(0,0)\}) \cup \{(1,1)\}$, and all but two diagonal circuits (Bamiloshin et al., 2023).

Restricting to $T^3$ (the canonical, classical TTT-matroid), the eight circuit-hyperplanes have their intersections governed entirely by the grid combinatorics of the affine plane.

3. Extension Properties and Representability Barriers

Three established extension properties draw sharp boundaries for matroid representability:

• Generalized Euclidean (GE): characterizes linear representability,
• Common-information (CI): characterizes folded-linear representability,
• Ingleton–Main (IM), Ahlswede–Körner (AK): characterize (almost)-entropic and algebraic representability (Bamiloshin et al., 2023).

In the TTT-matroid $T^3$, the flat pair $(A_{-1},A_1)$ fails to admit a CI-extension, so $T^3$ is neither folded-linear nor linear over any field. The dual $(T^3)^*$ fails AK and thus is not almost-entropic or algebraic. This analysis, based on explicit modular-cut and rank-count arguments, generalizes: none of the 181 TTT-matroids is folded-linear and none of their duals is algebraic.

The table below summarizes key representability features of TTT-matroids and their duals:

Matroid	Linear	Folded-linear	Almost-Entropic	Algebraic
$T^3$	No	No	Open	Open
$(T^3)^*$	No	No	No	No
Generic TTT	No	No	Open	Open
Dual TTT	No	No	No	No

For $T^3$ and generic TTT-matroids, algebraicity for characteristics $\neq 3$ remains open.

4. Pseudomodularity and Duality Phenomena

The lattice of flats of $T$ is pseudomodular in the Björner–Lovász sense: for every triple of flats $x, y, z$,

$$r(x\vee y\vee z)-r(x\vee y) = r(x\vee z)-r(x) = r(y\vee z)-r(y)$$

implies

$$r((x\vee z)\wedge(y\vee z)) - r(x\wedge y) = r(x\vee z)-r(x)$$

(Hochstättler, 13 Nov 2025).

Any violation would require three 5-point circuit-hyperplanes with pairwise intersections on three distinct colines, which is precluded by the sparse-paving structure. Thus, $T$ and all duals $M_k^*$ in the infinite family are pseudomodular.

Duality effects are dramatic:

• $(T^3)^*$ fails IM, hence is not algebraic.
• $(T^3)^*$ fails AK, so not almost-entropic. This yields infinite families of pseudomodular matroids whose duals are non-algebraic, and thus many novel excluded-minor candidates for both algebraic and almost-entropic matroids (Hochstättler, 13 Nov 2025, Bamiloshin et al., 2023).

5. Non-Algebraicity, Vámos Minors, and Infinite Generalization

Non-algebraicity of the primal $M_3$ and, dually, $(T^3)^*$ is established using the Ingleton–Main prism lemma. This lemma shows that in any algebraic matroid, three bounding lines of a prism must concur in a unique closure point. In $M_3$, specific "broken prism" configurations (triples such as $\{a_1,a_3\}, \{b_1,b_3\}, \{c_1,c_3\}$) lead to restrictions that embed a Vámos matroid minor (known to be non-algebraic), violating the closure of algebraic representability under minors (Hochstättler, 13 Nov 2025).

This construction extends to an infinite family: for each $k \ge 3$, the corresponding $M_k$ (rank-4 on $3k$ elements, defined analogously) is non-algebraic by producing Vámos minors using a general-prism argument, and their duals $M_k^*$ are pseudomodular sparse-paving matroids of rank $3k-4$.

6. Field Representability and Open Questions

Distinct members of the TTT-matroid family exhibit sharply contrasting representability behaviors:

• The maximal type I matroid $T_1^m$ is linearly representable only over fields of characteristic 3, admitting an explicit $5 \times 9$ matrix over $\mathbb{F}_3$.
• $T_1^3$ is representable over all fields.
• $T_2^m$ is not linearly representable (fails GE).

A computer-aided search via Frobenius flocks for characteristic 2 confirms that at least 62 of the 181 TTT-matroids are not algebraic over that characteristic; for the remainder and for $T^3$ specifically, algebraicity for characteristics $\neq 3$ remains unresolved (Bamiloshin et al., 2023).

The central open problem is: does $T^3$ (or any other TTT-matroid) admit an algebraic representation over some field of characteristic not equal to 3? A positive answer would give the first example of an algebraic, non-almost-entropic matroid; a negative answer would establish a large new excluded-minor family for the class of algebraic matroids.

7. Broader Significance and Implications

TTT-matroids, through their non-representability and duality behaviors, exemplify the limits of existing representability classes. The construction and structure of the TTT-matroid family deliver:

• A rich source of sparse-paving, non-linear, non-folded-linear matroids with algebraically "exotic" duals.
• Infinite sequences of pseudomodular matroids whose duals are non-algebraic, generalizing the classic $k=3$ TTT example to arbitrary $k \ge 3$.
• New excluded-minor candidates for the classes of folded-linear, (almost)-entropic, and algebraic matroids.

Their analysis—combining combinatorial, geometric, and lattice-theoretic techniques—has deepened understanding of non-representability and duality in matroid theory, and continues to stimulate open problems central to structural matroid theory (Hochstättler, 13 Nov 2025, Bamiloshin et al., 2023).