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. 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.

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.

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.

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.

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

1. 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. 2. 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.

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.

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

• Champernowne constant — 0.88
• Multi-index notation — 0.87
• Decimal representation — 0.87
• Bijective numeration — 0.87
• Bhargava factorial — 0.86

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