Arbitrarily Large Continuous Algebras on One Generator
Transactions of the American Mathematical Society
Generation of order-continuous algebras is investigated for various concepts of continuity. For the continuity understood as the preservation of joins of countably-directed sets, arbitrarily large infinitary continuous algebras on one generator are constructed. O. Introduction. Free continuous algebras have been explicitly described in  as algebras of trees for arbitrary (possibly infinitary) types of algebras and very general types of join-continuity, in a sense made precise in §1 below.
... ise in §1 below. Usually, in dealing with mathematical objects which are set-based, if free objects exist, then their existence can be proved using the time-honoured construction of universal objects (see Bourbaki  ) which depends on the formation of subobjects and products (which our algebras have), and on the property we call "bounded generation", that is, the cardinality of an object generated by a set X is bounded by some cardinal depending only on the cardinality of X. In this paper, we investigate the question of bounded generation for continuous algebras. We provide, in §2, a large variety of cases in which these algebras do indeed have bounded generation. Nevertheless, there is, as proved in §4, a class of continuous algebras which does not have bounded generation, namely those algebras with one operation of countable rank, where continuity refers to the existence and preservation of joins of countably-directed sets. A strengthening of this result, presented in §5, shows that the category of such algebras is not extremally co-wellpowered. To our knowledge, this is the only known example of an "algebraic" setting which does not have bounded generation but nevertheless has free objects. These results depend on set-theoretical assumptions which are discussed in §3. The construction, in §4, of a continuous algebra of power /3, on one generator, depends on the existence of a binary tree of depth /3 with leaves at all infinite levels. In §3, we give a characterization of those cardinals /3 for which such a tree exists; they include all cardinals which are below the first strongly inaccessible. In particular, if there are no strongly inaccessible cardinals, then we obtain unbounded generation. We do not know whether unbounded generation can be proved for these,