Poly-infix operators and operator families

Poly-infix operators and operator families are introduced as an alternative for working modulo associativity and the corresponding bracket deletion convention. Poly-infix operators represent the basic intuition of repetitively connecting an ordered sequence of entities with the same connecting primitive.

💡 Research Summary

The paper introduces the notion of poly‑infix operators and operator families as a formal alternative to the traditional handling of repeated infix operators through associativity and bracket omission. The authors observe that in elementary mathematics education and in programming, expressions such as “7+7+7” are routinely written without parentheses, relying on an implicit understanding that the binary “+” is associative. However, this reliance on associativity is not always justified, especially for non‑associative primitives, and it obscures the underlying syntactic structure.

To address this, the authors define a kernel Ψ (a binary function) and, for each positive integer n ≥ 2, an n‑ary operator Ψₙ : Sⁿ → S, where S is a given sort. The binary case Ψ₂ is identified with the ordinary infix expression x₁ Ψ x₂. Two recursive axiom schemes, AttLₙ₊₁ (association to the left) and AttRₙ₊₁ (association to the right), generate Ψₙ₊₁ from Ψₙ and Ψ₂:

• AttLₙ₊₁: Ψₙ₊₁(x₁,…,xₙ₊₁) = Ψₙ(Ψ₂(x₁,x₂),…,xₙ₊₁)
• AttRₙ₊₁: Ψₙ₊₁(x₁,…,xₙ₊₁) = Ψₙ(x₁,…,Ψ₂(xₙ,xₙ₊₁))

By induction these schemes yield the equalities
Ψₙ₊₁(x₁,…,xₙ₊₁) = Ψ₂(Ψₙ(x₁,…,xₙ),xₙ₊₁) = Ψ₂(x₁,Ψₙ(x₂,…,xₙ₊₁)).
Thus any n‑ary poly‑infix operator can be reduced to a nested application of the binary kernel, guaranteeing that the family of operators behaves exactly like an associative operator, but without having to invoke associativity as a meta‑property. Optional definitions Ψ₁(x)=x and Ψ₀=e_Ψ (the unit of Ψ) extend the framework to zero‑ary and unary cases.

The authors propose a poly‑infix notation: an expression x₁ Ψ x₂ … Ψ xₙ is simply written as Ψₙ(x₁,…,xₙ). This notation eliminates parentheses entirely; the AttL/AttR axioms ensure that any bracketing of the same expression yields the same term. For example,
x₁ Ψ x₂ Ψ x₃ Ψ x₄ Ψ x₅ = x₁ Ψ (x₂ Ψ x₃) Ψ x₄ Ψ x₅,
and any other placement of parentheses is provably equivalent.

The paper then supplies six concrete kernels to illustrate the breadth of the approach:

1. Pre‑arithmetic kernels – (i) parallel composition p₁‖…‖pₙ (process merging), (ii) sequential composition u₁;…;uₙ (instruction sequencing), and (iii) frame composition f₁⊕…⊕fₙ (graph‑like structure merging). These capture everyday combinatorial patterns in computer science and engineering.

2. Intra‑arithmetic kernels – addition (+) and multiplication (·). Repeated addition (e.g., 7+7+7) models multiplication, while repeated multiplication models exponentiation. The authors show how the poly‑infix view makes the pedagogical step from “adding a number many times” to “multiplying” completely formal.

3. Post‑arithmetic kernel – matrix multiplication for n‑dimensional square matrices over the rationals. Here the binary matrix product serves as the kernel, and the poly‑infix operator Ψₙ represents the product of n matrices without any parentheses.

Having established the syntax, the authors develop a poly‑infix specific equational logic. A Ψ‑expression is either a variable, a constant, a parenthesised non‑Ψ term, or an application of a poly‑infix operator. The length of a Ψ‑expression is the number of arguments supplied to its leading Ψ. A Ψ‑context C