The lambda-calculus is nominal alge- braic

In this paper we will write -[a 7→ -] as shorthand for (λa.-)-. Thus g[a 7→ h] stands for (λa.g)h and not for the term resulting from ‘substituting h for a in g’ (we write that as g[h/a], see Definition 44). The λ-calculus represents functions in programming languages [Pau96, Tho96], logic [Bar77, Lei94], theorem-provers [ABI96, Pau89], higher-order rewriting [BN98], and much more besides. However, the ‘λ’ in the λ-calculus has proved resistent to a treatment in universal algebra [BS81]. For example the property that “(λa.g)[b 7→ h] = λa.(g[b 7→ h]) when a does not occur free in h” cannot be represented in an algebraic framework, at least not obviously so, because of the freshness condition ‘a does not occur free in h’ which is necessary to avoid ‘accidental capture’ by λ. Similarly for the property “λa.(ga) = g when a does not occur free in g”. Nominal algebra is a form of universal algebra enriched with primitive constructs to handle names, binding, and freshness conditions — just like those that appear in informal specifications of the λ-calculus and other languages with binders. Nominal algebra has the feature that, thanks to the enriched constructs, it allows fully formal algebraic reasoning which