Positive Characteristic Set in Algebra and Learning

• A positive characteristic set is a finite ensemble defined by prime characteristics that uniquely distinguishes structures in algebra, combinatorics, and algorithmic learning.
• They exhibit structured behaviors such as arithmetic progressions and p-nested sequences, providing a robust framework for orbit intersections and matroid representations.
• In formal language learning, these sets act as telltale samples that enable efficient and polynomial-time identification of target languages.

A positive characteristic set is a finitely constructed set whose structure or existence is determined by the feature of prime characteristic in algebraic or combinatorial objects, or by the exclusivity of “positive” (i.e., affirming membership) examples in algorithmic learning. This concept manifests in several distinct mathematical domains, especially in algebraic dynamics, matroid theory, Diophantine combinatorics, and formal language learning, where the notion of “characteristic” is essential for both structural classification and algorithmic identification.

1. Definitions and Formal Properties

In the context of formal language theory, a positive characteristic set for a language $L_i$ (within a reference class $\mathcal{L}$) is a finite set $C_i \subseteq \Sigma^* \times \{1\}$ (i.e., pairs of strings with positive labels only) such that no other language $L_j \neq L_i$ in $\mathcal{L}$ can match all the positive evidence in $C_i$ without coinciding with $L_i$. Formally, $(C_i)_{i\in\mathbb{N}}$ is a family of positive characteristic sets if $C_i \subseteq L_i$, and for all $i \neq j$, if $C_i$ is consistent with $L_j$, then $L_j = L_i$ (Mousawi et al., 15 Nov 2025).

In algebraic dynamics and Diophantine geometry, “positive characteristic sets” refer to sets parameterizing orbit intersections, solution sets of linear recurrences, or realization sets of structures (such as matroids) over fields of positive characteristic. These sets exhibit combinatorial and arithmetic patterns—such as $p$-arithmetic and $p$-normal sets—imposed by the Frobenius endomorphism or characteristic $p > 0$-specific algebraic constraints (Ghioca, 2016, Rout, 2021, Cartwright et al., 2022).

2. Structural Results in Algebra and Dynamics

A central result in positive characteristic algebraic dynamics is the classification of sets $S = \{ n \in \mathbb{N}_{>0} : f^n(x) \in V(K) \}$, where $f$ is a self-map of the $N$-dimensional torus $T = \mathbb{G}_m^N$ over an algebraically closed field $K$ of characteristic $p > 0$, $V \subset T$ an irreducible curve, and $x \in T(K)$. The set $S$ is always a finite union of arithmetic progressions, finitely many $p$-arithmetic sequences of the form

$$A_{a,b,k} = \{ b + a p^{k n} : n \in \mathbb{Z}_{>0} \}$$

for $a, b \in \mathbb{Q}$, $k \in \mathbb{N}_{>0}$, and a finite exceptional set (Ghioca, 2016). No more exotic infinite structures can arise; this rigidity is a uniquely positive characteristic phenomenon and sharp, as $S$ can be infinite without containing any ordinary arithmetic progression.

In higher-dimensional orbit intersection problems, positive characteristic sets acquire a p-normal structure. For affine maps $\Phi_1, \Phi_2: K^d \to K^d$, the intersection set of two orbits

$$S = \{ (n_1, n_2) \in \mathbb{Z}^2 : \Phi_1^{n_1}(a_1) = \Phi_2^{n_2}(a_2) \}$$

is a finite union of translates of subgroups by singletons or elementary $p$-nested sets, with the order of $p$-nesting constrained by the dimension $d$. This class is stable under intersection and projection, reflecting the algebraic combinatorics induced by characteristic $p$ (Rout, 2021).

3. Characteristic Sets in Matroid Theory

Given a matroid $M$ with ground set $E$, one associates several characteristic sets according to representability:

• The linear characteristic set $\chi_L(M)$ comprises those characteristics $0$ or $p$ (prime) for which $M$ is linearly representable.
• The algebraic characteristic set $\chi_A(M)$ consists of characteristics where $M$ admits an algebraic realization (possibly over an extension).
• The Frobenius flock characteristic set $\chi_F(M)$ lists primes $p$ for which $M$ has a Frobenius flock representation.

Classical classification results (Rado–Vámos–Kahn) restrict linear characteristic sets to be either finite or cofinite (possibly including 0). Algebraic characteristic sets can be finite, cofinite, or sets of primes of arbitrary density in $[0,1]$ (e.g., sets built by congruence conditions), and every permissible combination of $\chi_L(M) \subseteq \chi_A(M)$ for finite or cofinite $C_L, C_A$ arises for some $M$ (Cartwright et al., 2022). Frobenius flock characteristic sets are always at least as large as algebraic characteristic sets, and frequently coincide with the full set of primes if $M$ is linearly realizable in characteristic zero or for duals.

4. Positive Characteristic Sets in Formal Language Learning

In algorithmic learning, positive characteristic sets provide the crucial sample restriction for learning from positive data only. For a class $\mathcal{L}$ of formal languages, these sets coincide (via a precise equivalence) with Angluin’s telltale sets—finite sets $T_i \subseteq L_i$ distinguishing $L_i$ with respect to all other $L_j \in \mathcal{L}$. A class has positive characteristic sets of polynomial size if and only if it admits polynomial-size telltales (Mousawi et al., 15 Nov 2025).

For relational pattern languages, concrete results include:

• The non-erasing equal-length class $\mathcal{L}_{len}$ for $|\Sigma| \geq 3$ admits linear-size positive characteristic sets, computable effectively in $O(|p|)$ time, where $p$ is the pattern length.
• For reversal-pattern languages over a binary alphabet, no positive characteristic sets exist in general, reflecting a non-learnability-from-positive-only-data barrier for this class.
• For certain restricted subclasses of binary equal-length patterns, small positive characteristic sets are again obtainable.

These properties yield efficient learning algorithms for families with positive characteristic sets of small size.

5. Methodologies and Canonical Examples

The construction and recognition of positive characteristic sets depend on the underlying algebraic or combinatorial framework:

• p-arithmetic sequences and $p$-nested sets arise by combining arithmetic progression structure with multiplicative $p$-power iteration. For example, $A_{a,b,k}$ encodes exponential spacing via Frobenius action (Ghioca, 2016).
• Elementary $p$-nested sets are defined as $$S_q(c_0; c_1, ..., c_k) = \{ c_0 + c_1 q^{f_1} + ... + c_k q^{f_k} : f_i \geq 0 \}$$, augmenting additive group structure with $p$-power indices (Rout, 2021).
• Matroid characteristic sets are characterized through model-theoretic embeddings and direct sum constructions; for arbitrary density sets, they induce families such as $\chi_A(M_q) = \{ p : p \not\equiv 1 \pmod q \}$, with density determined by Dirichlet’s theorem (Cartwright et al., 2022).
• Positive characteristic sets in learning are formed by exhaustively substituting patterns into minimal variable group configurations to guarantee identifiability, e.g., $\mathcal{S}_2(p, R)$ for equal-length patterns (Mousawi et al., 15 Nov 2025).

6. Applications, Limitations, and Open Problems

Positive characteristic sets govern the structure of solution sets in algebraic dynamics, determine the scope of learnability in algorithmic inference, and classify the representability of matroids. Their properties yield tight constraints—for instance, the impossibility of exotic infinite patterns in torsion point intersections, or the existence of learning-theoretic barriers for certain string relations.

Open problems include:

• Extending the polynomial-bound regime for positive characteristic sets in formal language classes (e.g., to more general patterns or relations).
• Understanding the complete landscape of algebraic and Frobenius flock representability for irregular matroids.
• Clarifying the full interaction between $p$-normal structures and higher-dimensional orbit intersection problems.
• Determining whether the logic of p-normal or p-arithmetic patterns fully captures all phenomena in explicit dynamical or combinatorial contexts in positive characteristic.

7. Comparative Table: Key Forms of Positive Characteristic Sets

Context	Canonical Set Form / Definition	Principal Reference
Algebraic dynamics (tori, curves)	Union of arithmetic & $p$-arithmetic sequences	(Ghioca, 2016)
Orbit intersection (linear/toric)	Finite union of subgroup translates by $p$-nested sets	(Rout, 2021)
Matroid representability	(Linear/Algebraic/Frobenius) characteristic sets	(Cartwright et al., 2022)
Learning formal languages	Positive characteristic sets / telltales	(Mousawi et al., 15 Nov 2025)

Each paradigm exploits the arithmetic and combinatorial consequences of working in positive characteristic, yielding structural phenomena unattainable in characteristic zero.