Closure functions and general iterates as reflectors (Q808275)

In a previous paper [Ann. Pure Appl. Logic 40, No.3, 217-256 (1988; Zbl 0664.68047)] the authors have studied primitive iterates, corresponding to for-until-do statements; in the present paper they deal with general iterates, corresponding to while-do statements. An analogy between closure functions and general iterates leads the authors to investigate their common algebraic structure. To this aim they develop a closure theory for categories from which both the classical closure theory and the closure theory for iteration can be obtained as particular cases. Both the closure functions and the general iterates are the (object part of) reflector, i.e. left adjoints for the inclusion functor. This result supports the asserted importance of adjoints in computer science. Furthermore, the offered semantics of the while-do statement is independent from the semantics of recursive procedures and closer to the usual iterative implementation of the construct.