Lunar arithmetic¶

Replace digit addition by maximum and digit multiplication by minimum, extending them positionally without carries so nonnegative base-b numerals form a closed idempotent arithmetic with altered sums, products, factors, and primes.

Version: v2 · 2026-08-30 · History
Domain-specific #: 2216
Origin domain: recreational number theory
Subdomain: carryless idempotent arithmetic

Core Idea¶

In base \(b\), lunar arithmetic defines digit addition by \(a\oplus c=\max(a,c)\) and digit multiplication by \(a\otimes c=\min(a,c)\); multidigit addition is columnwise and multiplication combines shifted digitwise partial products using \(\oplus\), with no carries.^[1] Maximum selects the larger digit in each aligned column, minimum forms each digit product, and maximum superposes the shifted partial products, so both operations remain within the digit alphabet and inherit commutative, associative, idempotent, and distributive behavior from the max-min lattice.

Its autonomous residual is the paired max-min positional arithmetic and its carryless extension, not ordinary carryless addition alone, tropical arithmetic, Boolean arithmetic, a typographical joke, or arithmetic performed on a lunar calendar. The identity fails when ordinary carries are introduced, minimum and maximum are exchanged without notice, numeral base changes mid-calculation, ordinary multiplication tables are reused, or familiar definitions such as prime and factorial retain their usual identities instead of the lunar ones.

Recognition requires an analyst to declare the base and symbols, align digits explicitly, apply max for addition and min for each digit product, combine partial rows by max without carrying, and recheck any familiar arithmetic term under the lunar identities. Once established, it supports studying an idempotent semiring-like arithmetic, generating lunar primes and powers, linking binary lunar products to sumsets, testing which familiar number-theoretic notions survive changed operations, and illustrating dependence on operational rules without turning those uses into the definition.

Structural Signature¶

Carrier: nonnegative integers represented as finite digit strings in a fixed positional base, with leading-zero equivalence and digitwise maximum and minimum operations
Inputs or antecedent state: base, digit alphabet, numeral representation, alignment by place value, maximum addition, minimum digit multiplication, carryless partial-product shifts, and the induced identities and factorization convention
Constitutive operation: Maximum selects the larger digit in each aligned column, minimum forms each digit product, and maximum superposes the shifted partial products, so both operations remain within the digit alphabet and inherit commutative, associative, idempotent, and distributive behavior from the max-min lattice
Invariant: a positional base is fixed, the two digit operations are maximum and minimum, multidigit rules use place alignment without ordinary carries, and every claimed sum, product, identity, or factorization is interpreted under those same operations
Recognition test: declare the base and symbols, align digits explicitly, apply max for addition and min for each digit product, combine partial rows by max without carrying, and recheck any familiar arithmetic term under the lunar identities
Output or consequence: studying an idempotent semiring-like arithmetic, generating lunar primes and powers, linking binary lunar products to sumsets, testing which familiar number-theoretic notions survive changed operations, and illustrating dependence on operational rules
Failure boundary: ordinary carries are introduced, minimum and maximum are exchanged without notice, numeral base changes mid-calculation, ordinary multiplication tables are reused, or familiar definitions such as prime and factorial retain their usual identities instead of the lunar ones

What It Is Not¶

It is not the whole field of recreational number theory; many objects in that field do not satisfy its constitutive rule.
It is not its canonical example. In decimal lunar arithmetic, \(976\oplus348=978\) because the aligned columns take maxima \(9,7,8\); at the digit level \(2\otimes7=2\) because the product is their minimum. That is an instance, not a definition.
It is not Tropical Arithmetic. Tropical semirings usually combine minimum or maximum with ordinary addition. Lunar arithmetic instead uses max as addition and min as digit multiplication, then extends those digit rules through positional carryless multiplication.
It is not an unrestricted metaphor. The largest digit \(b-1\) is the multiplicative identity and zero is the additive identity, so the definition of a lunar prime uses \(b-1\) where ordinary arithmetic uses one; leading zeros must not create spurious factors

Scope of Application¶

Lunar arithmetic applies when the analyst can specify nonnegative integers represented as finite digit strings in a fixed positional base, with leading-zero equivalence and digitwise maximum and minimum operations and establish that a positional base is fixed, the two digit operations are maximum and minimum, multidigit rules use place alignment without ordinary carries, and every claimed sum, product, identity, or factorization is interpreted under those same operations. The entry covers the Applegate–LeBrun–Sloane max-min arithmetic on nonnegative positional numerals; signed, fractional, infinite, or alternative digit systems need additional definitions.^[2]

Recognition. declare the base and symbols, align digits explicitly, apply max for addition and min for each digit product, combine partial rows by max without carrying, and recheck any familiar arithmetic term under the lunar identities
Comparison. Compare legitimate instances through base, digit order, leading zeros, alignment, addition rule, multiplication rule, identities, idempotence, distributivity, factorization, prime convention, encoded set interpretation, and notation.
Boundary. The largest digit \(b-1\) is the multiplicative identity and zero is the additive identity, so the definition of a lunar prime uses \(b-1\) where ordinary arithmetic uses one; leading zeros must not create spurious factors
Use. Preserve every assumption when using the identity for studying an idempotent semiring-like arithmetic, generating lunar primes and powers, linking binary lunar products to sumsets, testing which familiar number-theoretic notions survive changed operations, and illustrating dependence on operational rules.

Clarity¶

A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because the earlier name dismal arithmetic and the later name lunar arithmetic refer to the same system, while carryless and tropical arithmetic can misleadingly suggest different operation tables. The disciplined statement is that the object counts as Lunar arithmetic exactly when a positional base is fixed, the two digit operations are maximum and minimum, multidigit rules use place alignment without ordinary carries, and every claimed sum, product, identity, or factorization is interpreted under those same operations

Identity and measurement remain separate. Calculations are exact once base and digit rules are fixed; software tests should verify alignment, leading-zero normalization, and operation tables rather than compare results with ordinary arithmetic. Approximation or noisy evidence may weaken a classification without changing its definition.

Manages Complexity¶

The abstraction compresses binary and higher-base forms, polynomial interpretations, lunar primes and powers, standard-sequence analogues, sumset encodings, magic-square constructions, and alternative carryless systems into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.

Compression can hide assumptions. A responsible use therefore declares base, digit order, leading zeros, alignment, addition rule, multiplication rule, identities, idempotence, distributivity, factorization, prime convention, encoded set interpretation, and notation and returns to the full diagnostic whenever a convention or boundary case changes.

Abstract Reasoning¶

Type the carrier. Establish nonnegative integers represented as finite digit strings in a fixed positional base, with leading-zero equivalence and digitwise maximum and minimum operations and reject examples from a different problem.
Lock the rule. Express that a positional base is fixed, the two digit operations are maximum and minimum, multidigit rules use place alignment without ordinary carries, and every claimed sum, product, identity, or factorization is interpreted under those same operations independently of one notation or implementation.
Derive carefully. Infer studying an idempotent semiring-like arithmetic, generating lunar primes and powers, linking binary lunar products to sumsets, testing which familiar number-theoretic notions survive changed operations, and illustrating dependence on operational rules only under the stated assumptions.
Stress-test. Contrast the legitimate boundary case—The largest digit \(b-1\) is the multiplicative identity and zero is the additive identity, so the definition of a lunar prime uses \(b-1\) where ordinary arithmetic uses one; leading zeros must not create spurious factors—with this counterexample: decimal addition performed without carries but using digit sums modulo ten is not lunar addition because lunar addition chooses the maximum digit rather than a residue.

Knowledge Transfer¶

Transfer within recreational number theory is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from In decimal lunar arithmetic, \(976\oplus348=978\) because the aligned columns take maxima \(9,7,8\); at the digit level \(2\otimes7=2\) because the product is their minimum. to For binary numerals encoding finite subsets of nonnegative integers by their occupied positions, lunar multiplication corresponds to forming the sumset of the represented exponent sets. demonstrates that continuity.^[3]

Outside the domain, only the skeleton—replace familiar local operations by an ordered pair of lattice operations and propagate those rules compositionally through a representation—travels automatically. The terms digit, positional base, maximum, minimum, carryless, idempotent, semiring, additive identity, multiplicative identity, factor, lunar prime, and sumset retain domain-specific meanings, so every role and inference must be revalidated.

Examples¶

Canonical¶

In decimal lunar arithmetic, \(976\oplus348=978\) because the aligned columns take maxima \(9,7,8\); at the digit level \(2\otimes7=2\) because the product is their minimum. No column creates or receives a carry, and the examples reveal that the symbols plus and times name different operations from ordinary integer arithmetic. It is canonical because the carrier, rule, invariant, and consequence are all inspectable.^[1]

Mapped back: nonnegative integers represented as finite digit strings in a fixed positional base, with leading-zero equivalence and digitwise maximum and minimum operations → Maximum selects the larger digit in each aligned column, minimum forms each digit product, and maximum superposes the shifted partial products, so both operations remain within the digit alphabet and inherit commutative, associative, idempotent, and distributive behavior from the max-min lattice → a positional base is fixed, the two digit operations are maximum and minimum, multidigit rules use place alignment without ordinary carries, and every claimed sum, product, identity, or factorization is interpreted under those same operations → studying an idempotent semiring-like arithmetic, generating lunar primes and powers, linking binary lunar products to sumsets, testing which familiar number-theoretic notions survive changed operations, and illustrating dependence on operational rules

Applied / In Practice¶

For binary numerals encoding finite subsets of nonnegative integers by their occupied positions, lunar multiplication corresponds to forming the sumset of the represented exponent sets. Minimum acts as Boolean conjunction on binary digit products and maximum as disjunction across equal exponent positions, so shifted partial products record every pairwise exponent sum. It qualifies only after the same diagnostic and failure boundary are checked.^[2]

Mapped back: declared instance → recognition test → boundary check → qualified use

Structural Tensions¶

T1: Exact identity vs. practical recognition. The constitutive condition may be exact while evidence is indirect. Diagnostic: Can the reviewer state both the condition and the warrant?
T2: Canonical form vs. variants. binary and higher-base forms, polynomial interpretations, lunar primes and powers, standard-sequence analogues, sumset encodings, magic-square constructions, and alternative carryless systems can preserve or change the identity. Diagnostic: Which named role is invariant across the variants?
T3: Compression vs. hidden assumptions. The label is useful only while prerequisites remain visible. Diagnostic: Can each downstream inference be traced to a declared assumption?
T4: Autonomy vs. reduction. The candidate uses broader structures but claims the paired max-min positional arithmetic and its carryless extension, not ordinary carryless addition alone, tropical arithmetic, Boolean arithmetic, a typographical joke, or arithmetic performed on a lunar calendar. Diagnostic: Does that residual still support independent recognition after the parent and neighbors are subtracted?

Structural–Framed Character¶

The entry is structurally mixed but domain-framed. Its portable skeleton is replace familiar local operations by an ordered pair of lattice operations and propagate those rules compositionally through a representation; its identity-bearing terms are digit, positional base, maximum, minimum, carryless, idempotent, semiring, additive identity, multiplicative identity, factor, lunar prime, and sumset. Those terms determine admissible objects, evidence, and consequences inside recreational number theory.

Structural Core vs. Domain Accent¶

The structural core is a carrier governed by Maximum selects the larger digit in each aligned column, minimum forms each digit product, and maximum superposes the shifted partial products, so both operations remain within the digit alphabet and inherit commutative, associative, idempotent, and distributive behavior from the max-min lattice and tested by declare the base and symbols, align digits explicitly, apply max for addition and min for each digit product, combine partial rows by max without carrying, and recheck any familiar arithmetic term under the lunar identities. The domain accent is constitutive rather than decorative, so an analogy that preserves only the skeleton is not another instance of Lunar arithmetic.

The proposed strict upward parent is prime:closure. The fixed digit carrier and induced nonnegative numerals are literally closed under the designated maximum-addition and minimum-based multiplication operations; their positional, idempotent arithmetic supplies the specialization. The edge is proposal-only and points to a frozen prior-baseline Prime.

The entry does not collapse into the parent because the paired max-min positional arithmetic and its carryless extension, not ordinary carryless addition alone, tropical arithmetic, Boolean arithmetic, a typographical joke, or arithmetic performed on a lunar calendar A thematic neighbor is declined whenever it does not literally subsume that rule.

The prospective workspace queue contains one strict upward edge to prime:closure. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction Lunar arithmetic Domain-specific

Parents (1) — more general patterns this builds on

Lunar arithmetic is a kind of Closure Prime

The proposed strict upward parent is prime:closure.

Hierarchy path (1) — routes to 1 parentless root

Lunar arithmetic → Closure

Neighborhood in Abstraction Space¶

Lunar arithmetic sits in a moderately populated region (58^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Numeration & Arithmetic Representations (15 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Carryless arithmetic. A broader family that can use modular digit sums and products rather than maximum and minimum.
Tropical semiring. Usually pairs min or max with ordinary addition and has different identities and positional semantics.
Boolean semiring. On two truth values it resembles the base-two digit operations, but lunar arithmetic extends them to positional numerals and arbitrary bases.
Ordinary prime. Uses ordinary multiplication and unit one; a lunar prime is irreducible under lunar multiplication with the largest digit as unit.

References¶

[1] David Applegate, Marc LeBrun, and N. J. A. Sloane, 'Dismal Arithmetic,' Journal of Integer Sequences 14 (2011), arXiv:1107.1130. registry ↩a ↩b

[2] Charles Woll, 'There Is a 3×3 Magic Square of Squares on the Moon—A Lot of Them, Actually,' The Mathematical Intelligencer 41, 73–76 (2019), DOI 10.1007/s00283-018-09866-4. registry ↩a ↩b

[3] Gal Gross, 'Maximally Additively Reducible Subsets of the Integers,' Journal of Integer Sequences 23, Article 20.10.5 (2020), arXiv:1908.05220. registry ↩