On some developments in the representation theory of cylindric-like algebras

In this survey paper we recall the representation problem of certain classes of algebras of relations. Two specific recent research directions in algebraic logic are surveyed: the one is connected with the representation theory of cylindric relativized set algebras including Resek’s representation theorem and merry-go-round axioms, the other discusses the representation problem in some non-well-founded set theories where the axiom of foundation is replaced by Boffa’s weak anti-foundation axiom. This paper is dedicated to Walter Taylor. Received September 8, 2005; accepted in final form January 12, 2006. The first author was supported by Hungarian National Foundation for Scientific Research grants T43242 and T035192. The second author was supported by Hungarian National Foundation for Scientific Research grants D042177, T43242 and T035192.