[Paper Review] Poly-infix operators and operator families

J.A. Bergstra, Alban Ponse|arXiv (Cornell University)|May 5, 2015

Advanced Topics in Algebra11 references3 citations

TL;DR

This paper introduces poly-infix operators and operator families as a formal alternative to standard infix notation with implicit associativity, proposing that repeated unbracketed uses of an infix operator (e.g., 2+2+2) should be treated as a single n-ary operator rather than nested binary operations. The approach provides a dedicated equational logic for such expressions, eliminating reliance on bracketing conventions and offering a syntactically and semantically coherent foundation for arithmetic education and algebraic reasoning.

ABSTRACT

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.

Motivation & Objective

• To provide a formal foundation for the use of unbracketed repeated infix operators (e.g., 2+2+2) as first-class n-ary operators, rather than as nested binary operations.
• To resolve ambiguities in elementary arithmetic education where expressions like 2+2+2+2 are conventionally interpreted via associativity but lack explicit syntactic treatment.
• To support an expression-oriented view of arithmetic that treats multi-operand expressions as primitive, distinct from value-oriented interpretations.
• To enable a more coherent and pedagogically sound equational logic for reasoning about repeated operations without relying on bracketing conventions.
• To support the development of new reference levels in arithmetical competence for special education and remedial teaching by formalizing intuitive arithmetic practices.

Proposed method

• Introduces poly-infix operator families defined by a kernel operator Ψ, with n-ary operators Ψₙ: Sⁿ → S for n ≥ 2.
• Defines two axioms—(AttL n+1) and (AttR n+1)—to formalize left and right association, ensuring that all bracketings are derivable from the base form.
• Uses poly-infix notation: x₁ Ψ x₂ … xₙ₊₁ to denote Ψₙ₊₁(x₁, ..., xₙ₊₁), treating the entire sequence as a single operator instance.
• Establishes that associativity follows from the axioms, and that any bracketing of a sequence can be derived using the axioms.
• Applies the framework to pre-arithmetical (e.g., parallel/sequential composition) and intra-arithmetical (e.g., addition, multiplication) operator kernels.
• Proposes a dedicated equational logic for reasoning about expressions like 2+2+2+2, treating them as single five-place operations rather than nested binary operations.

Experimental results

Research questions

• RQ1 How can repeated unbracketed infix operations (e.g., 2+2+2) be formally treated as a single n-ary operator rather than as nested binary operations?
• RQ2 What axiomatic system is required to derive all possible bracketings of a poly-infix expression while preserving syntactic and semantic coherence?
• RQ3 In what ways does treating repeated infix operations as poly-infix operators improve clarity and consistency in elementary arithmetic education?
• RQ4 How does the poly-infix approach support a formal distinction between expression-oriented and value-oriented views of arithmetic?
• RQ5 What are the implications of this framework for designing new reference levels of arithmetical competence in special education?

Key findings

• The poly-infix operator family formalism allows expressions like 2+2+2+2 to be treated as a single five-place operator, eliminating the need for implicit bracketing or associativity assumptions.
• The axioms (AttL n+1) and (AttR n+1) are sufficient to derive all possible bracketings of a poly-infix expression, ensuring full associativity without relying on external conventions.
• The framework enables a direct equational proof of identities such as 2+2+2+2+2 = 2+2+4+2, using a dedicated proof rule based on the poly-infix structure.
• The approach provides a syntactic foundation for educational practices such as those in Dutch primary education, where repeated addition is taught without explicit brackets.
• The method supports a coherent expression-oriented view of arithmetic, reducing inconsistencies that arise when mixing expression-based and value-based interpretations of fractions or sums.
• The framework is consistent with proposals in [12] for new reference levels in arithmetical competence, offering a formal basis for remedial and special education curricula.