On the poset of computation rules for nonassociative calculus

The symmetric maximum, denoted by v, is an extension of the usual max operation so that 0 is the neutral element, and -x is the symmetric (or inverse) of x, i.e., x v(-x)=0. However, such an extension does not preserve the associativity of max. This fact asks for systematic ways of parenthesing (or bracketing) terms of a sequence (with more than two arguments) when using such an extended maximum. We refer to such systematic (predefined) ways of parenthesing as computation rules. As it turns out there are infinitely many computation rules each of which corresponding to a systematic way of bracketing arguments of sequences. Essentially, computation rules reduce to deleting terms of sequences based on the condition x v(-x)=0. This observation gives raise to a quasi-order on the set of such computation rules: say that rule 1 is below rule 2 if for all sequences of numbers, rule 1 deletes more terms in the sequence than rule 2. In this paper we present a study of this quasi-ordering of computation rules. In particular, we show that the induced poset of all equivalence classes of computation rules is uncountably infinite, has infinitely many maximal elements, has infinitely many atoms, and it embeds the powerset of natural numbers ordered by inclusion.

💡 Research Summary

The paper investigates the “symmetric maximum” operation, denoted ⊙, which extends the ordinary max to the whole set of integers by making 0 the neutral element and defining a ⊙ (‑a) = 0 for every integer a. Apart from this symmetry, ⊙ coincides with the usual maximum on positive numbers and with the minimum on negative numbers; in mixed sign cases it returns the operand with the larger absolute value. This extension inevitably breaks associativity: for example (‑3)⊙(3⊙2)=0 while ((‑3)⊙3)⊙2=2. Consequently, evaluating expressions of the form a₁⊙a₂⊙…⊙aₙ requires a systematic way of inserting parentheses so that the result is unambiguous.

The authors formalize such systematic parenthesizations as “computation rules”. The key observation is that non‑associativity occurs precisely when a subsequence contains a pair of opposite numbers whose symmetric maximum is 0. A computation rule therefore works by deleting elements from a sequence until the remaining sequence satisfies the associativity condition (either it has length ≤2 or the total symmetric maximum of the whole sequence equals the symmetric maximum of its negated counterpart).

To encode sequences, the authors first reorder the terms by decreasing absolute value and then record, for each distinct absolute value nₖ, the multiplicities (pₖ, mₖ) of the positive and negative occurrences. The whole sequence is thus represented by the list ψ(σ)=((p₁,m₁),…,(p_q,m_q)). Non‑associativity is exactly the situation p₁>0 and m₁>0.

Five elementary transformation rules ρ₁,…,ρ₅ are introduced:

• ρ₁: If p₁>1 and m₁>0, reduce p₁ to 1 (keep a single positive copy of the largest absolute value).
• ρ₂: If m₁>1 and p₁>0, reduce m₁ to 1 (keep a single negative copy).
• ρ₃: If both p₁ and m₁ are positive, delete min(p₁,m₁) symmetric pairs; if a pair becomes (0,0) it is removed and the remaining list is renumbered.
• ρ₄: If p₂>0 (second‑largest absolute value has a positive occurrence), set p₂ to 0 (delete all such positives).
• ρ₅: Analogously, if m₂>0, set m₂ to 0.

Each rule acts only when its pre‑condition holds; otherwise it is the identity. A computation rule is any finite or infinite word over the alphabet {ρ₁,…,ρ₅}. A rule is called well‑formed (w.f.c.r.) if, for every input sequence σ, the transformed sequence R(σ) lies outside the non‑associative set S₀, i.e., the rule always succeeds in making the sequence associative.

The authors then define a quasi‑order on the set of computation rules: R₁ ≤ R₂ iff for every σ the number of deletions performed by R₁ is at least that performed by R₂. This relation is not antisymmetric, so they factor by the equivalence relation R₁ ∼ R₂ (identical final result for all σ) and study the induced partially ordered set (poset) of equivalence classes.

The main structural results are:

1. Uncountability – By associating to each subset A⊆ℕ a rule R_A that deletes exactly the levels indexed by A, they embed the power set of ℕ (ordered by inclusion) into the poset. Hence the poset has cardinality 2^{ℵ₀}.

2. Infinitely many atoms – An atom is a minimal non‑zero element of the poset. For each absolute‑value level k, the rule that deletes a single pair at that level (and nothing else) is an atom. Since there are infinitely many levels, there are infinitely many atoms.

3. Infinitely many maximal elements – A maximal element is a rule that cannot be extended by further deletions without becoming equivalent to the universal “delete everything” rule. By fixing any infinite set of levels on which deletions are performed and leaving the rest untouched, one obtains a distinct maximal element. Thus there are infinitely many.

4. Embedding of the powerset – The mapping A ↦