Core Concepts

The core message of this article is that there is a one-to-one correspondence between composite theories and distributive laws between monads. The authors provide a full and self-contained proof of this correspondence, and also give criteria for identifying a minimal set of distribution axioms that axiomatize a composite theory.

Abstract

The article explores the correspondence between composite theories and distributive laws between monads. It makes the following key points:
Composite theories are the algebraic equivalent of distributive laws between monads. A composite theory U contains the function symbols and equations of two algebraic theories S and T, as well as a set of distribution axioms that specify how equality of mixed terms can be reduced to equality in S and T.
The authors provide a full and self-contained proof of the correspondence between composite theories U and distributive laws λ: ST → T S, where S and T are the corresponding finitary Set-monads.
Section 4 shows how to construct a distributive law from a given composite theory, while Section 5 shows how to construct a composite theory from a given distributive law, using term rewriting techniques.
The authors introduce the concept of functorial rewriting systems to reason about strings of functors and obtain a separation of U-terms. They prove that the functorial rewriting system used in the construction is terminating and confluent-commuting.
In Section 6, the authors give criteria that ensure a certain minimal set of distribution axioms E' ⊆ Eλ suffices to axiomatize the composite theory U. They prove that if the term rewriting system corresponding to E' is terminating, then ES ∪ ET ∪ E' axiomatizes U.
The authors apply their results to establish presentations of some composite monads/theories, such as the theory of rings.

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Deeper Inquiries

The criteria for identifying a minimal set of distribution axioms can be extended or generalized to other types of composite theories by considering the structural properties of the theories involved. One approach could be to analyze the interaction between the different components of the composite theory and determine the essential distribution axioms that capture these interactions. By examining the algebraic structure of the theories and the relationships between their operations, it may be possible to identify common patterns or principles that lead to a minimal set of distribution axioms. Additionally, exploring the properties of the monads associated with the theories and their categorical interpretations could provide insights into generalizing the criteria for identifying minimal distribution axioms in composite theories.

The correspondence between composite theories and distributive laws has various potential applications and implications beyond the examples provided in the paper. One significant application is in the field of programming languages and semantics, where monads and algebraic theories are used to model computational effects. By understanding the relationship between composite theories and distributive laws, researchers can develop more efficient and expressive programming languages that incorporate multiple computational effects. This can lead to advancements in areas such as functional programming, type theory, and program verification.
Furthermore, the correspondence between composite theories and distributive laws can also be applied in the study of abstract algebra, category theory, and mathematical logic. It provides a framework for analyzing the structure of composite algebraic theories and understanding the connections between different mathematical structures. This can lead to new insights into the relationships between various algebraic systems and provide a deeper understanding of the underlying principles governing these structures.

The term rewriting techniques used in this paper have connections to various areas of mathematics and computer science, including higher-order rewriting and categorical semantics. In higher-order rewriting, the focus is on rewriting rules that manipulate higher-order terms, functions, or predicates. The techniques employed in the paper, such as defining rewrite rules for composite theories and analyzing the termination and confluence properties of rewriting systems, can be extended to higher-order rewriting systems.
In categorical semantics, the correspondence between composite theories and distributive laws can be viewed through the lens of category theory. The use of natural transformations, monads, and algebraic theories in the paper aligns with the categorical approach to modeling mathematical structures and relationships. By exploring the connections between term rewriting in algebraic theories and categorical semantics, researchers can bridge the gap between syntactic manipulation of terms and semantic interpretations in category theory. This integration can lead to a deeper understanding of the relationships between algebraic structures and their categorical representations.

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