Nominal Algebra and the HSP Theorem

Abstract

Nominal algebra is a logic of equality developed to reason algebraically in the presence of binding. In previous work it has been shown how nominal algebra can be used to specify and reason algebraically about systems with binding, such as first-order logic, the lambda-calculus, or process calculi. Nominal algebra has a semantics in nominal sets (sets with a finitely-supported permutation action); previous work proved soundness and completeness. The HSP theorem characterises the class of models of an algebraic theory as a class closed under homomorphic images, subalgebras, and products, and is a fundamental result of universal algebra. It is not obvious that nominal algebra should satisfy the HSP theorem: nominal algebra axioms are subject to so-called freshness conditions which give them some flavour of implication; nominal sets have significantly richer structure than the sets semantics traditionally used in universal algebra. The usual method of proof for the HSP theorem does not obviously transfer to the nominal algebra setting. In this paper we give the constructions which show that, after all, a 'nominal' version of the HSP theorem holds for nominal algebra; it corresponds to closure under homomorphic images, subalgebras, products, and an atoms-abstraction construction specific to nominal-style semantics.

On universal algebra over nominal sets

“…We prove that these translations are semantically invariant and we obtain an HSPA theorem for nominal equational logic and a new proof of the HSPA theorem of nominal algebra (Gabbay, 2008).…”

“…This paper describes a step towards universal algebra over nominal sets. There has been some work in this direction, most notably by M.J. Gabbay (Gabbay, 2008). The originality of our approach is that we do not start from the analogy between sets and nominal sets.…”

“…The originality of our approach is that we do not start from the analogy between sets and nominal sets. As shown in (Gabbay, 2008), this is possible, but it requires ingenuity and ad hoc constructions. For example, the logic of (Gabbay, 2008) is not standard equational logic and even fundamental notions such as variables and free algebras have to be revisited.…”

“…(Adámek et al, 1990, Exercise 16G)), it is straightforward to transfer it to the nominal setting: a class ofL-algebras is equationally definable if and only if it is closed under products, subalgebras and presheaf-quotients. The usefulness of changing the perspective from Nom to Set I is highlighted by Gabbay's remark in (Gabbay, 2008) that "An attempt to directly transfer proofs of the HSP theorem to the nominal setting fails". Section 4 extracts from the abstract category theoretic treatment of Section 3 a standard equational calculus.…”

“…We therefore introduce a fragment of our equational logic, called the uniform fragment. We prove an HSPA theorem in the style of (Gabbay, 2008): a class ofL-algebras is definable by uniform equations if and only if it is closed under quotients, subalgebras, products and abstraction. This shows that the uniform fragment of equational logic has the same expressiveness as the nominal algebra of (Gabbay, 2008).…”