Abstract
In this work we present a novel formalization of universal algebra in Agda. We show that heterogeneous signatures can be elegantly modelled in type-theory using sets indexed by arities to represent operations. We prove elementary results of heterogeneous algebras, including the proof that the term algebra is initial and the proofs of the three isomorphism theorems. We further formalize equational theory and prove soundness and completeness. At the end, we define (derived) signature morphisms, from which we get the contravariant functor between algebras; moreover, we also proved that, under some restrictions, the translation of a theory induces a contra-variant functor between models.
Highlights
Universal algebra [2] is the study of different types of algebraic structures at an abstract level, revealing common results which are valid for all of them and allowing for a unified definition of constructions
We showed that the translations of theories arising from derived signature morphisms induces a contra-variant functor between models
As far as we know, heterogeneous universal algebra has not attracted a great interest in the academic community of type theory
Summary
Universal algebra [2] is the study of different types of algebraic structures at an abstract level, revealing common results which are valid for all of them and allowing for a unified definition of constructions (for example, products, subalgebra, congruences). Gunther et al / Electronic Notes in Theoretical Computer Science 338 (2018) 147–166 available formalizations in type theory (which we discuss in the conclusion) This situation is to be contrasted with impressive advances in mechanization of particular algebraic structures as witnessed, for example, by the proof of the Feit-Thompson theorem in Coq by Gonthier and his team [21]. In this work we present an Agda library of multi-sorted universal algebra aiming both a reader with a background in the area of algebraic specifications and the community of type theory. For the former, we try to explain enough Agda in order to keep the paper self-contained; we will recall the most important definitions of universal algebra.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
