On the Algebraic Structure of Rooted Trees

Generalizing Substitution

“…In Aczel et al's [2] terminology (which builds on that of Elgot et al [8,9]), their monad is completely iterative. Definition 2.7.…”

“…The study of the structure of the algebras of rational and non-wellfounded H-terms was started by Elgot and colleagues [8,9] in the universal-algebra setting, where C = Set and H is polynomial.…”

“…Manes [14]. In contrast, the comparably fundamental fact that a similar monad, which even enjoys the additional property of having iterations for all guarded substitution rules (complete iterativeness), arises from the inverses of the final (A+H−)-coalgebras (if existing), i.e., the algebras of non-wellfounded H-terms over different A, was only recently pointed out by Moss [17] and Aczel, Adámek, Milius and Velebil [1,2] (their substitution and solution theorems), although the "down-to-earth" universal-algebra case, where C = Set and H is a polynomial endofunctor, was settled in the 1970s by Elgot and colleagues [9].…”

Rational term rewriting

“…As far as we know, Ginali in her Ph.D. thesis (see [17]) and independently Elgot, Bloom and Tindell [14] were the first to prove a correspondence result between the class of regular trees and Elgot's free iterative theories. Building on that result, Bloom and l~sik proved in [3] the following theorem.…”

A Description of Iterative Reflections of Monads (Extended Abstract)

“…So far, iterative theories were considered over arbitrary signatures [9,10] or arbitrary endofunctors [3] but without studying the effect of equational laws on given operations. For example, Elgot et al described in [10] the free iterative theory on a signature Σ as the theory R Σ of all rational Σ-trees (that is, Σ-trees with only finitely many subtrees up to isomorphism). The free iterative theory can be thought of as the closure of the theory formed by all Σ-terms under unique solutions of recursive equations.…”