Number & Formal Language Properties

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Abstractions that define special classes of numbers, operations, and formal languages, including numeric classifications (Achilles number, sexy primes, dyadic rational), degenerate or foundational operations (empty sum, ternary operation), and formal-language constructions (Dyck language, quotient of a formal language).

7 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Achilles Number — A positive integer that is powerful because every prime exponent is at least two, yet is not a perfect power because those exponents have greatest common divisor one.
• Dyadic Rational — A rational number expressible as an integer divided by a power of two, equivalently a rational number with a terminating binary expansion.
• Dyck language — The context-free language of bracket strings whose openings and closings are correctly balanced, ordered, and nested, with type-correct matching where several bracket pairs exist.
• Empty Sum — A summation with no terms, assigned the additive identity of its carrier.
• Quotient of a Formal Language — An operation on formal languages that removes a suffix or prefix drawn from another language: L₁/L₂ contains w when wx∈L₁ for some x∈L₂, with a dual left quotient defined by xw∈L₁.
• Sexy Primes — A pair of prime numbers separated by exactly six, with triplet and higher variants requiring repeated six-unit prime gaps.
• Ternary Operation — A total three-input function—internally T:A³→A when defined on a set—that maps each ordered triple to one output, with any further identities stated separately.