Zur Existenz freier Algebren einer r-dimensionalen Theorie

Abstract

In [2] Linton introduced the notion of theory in such a way that he was able to indicate explicitly the leftadjoint of the underlying functor from the category of algebras over the theory in the sets. His method defines, in the case of the restriction on a r-ary theory [2] or a algebraische Theorie der Dimension r [3] (r an infinite cardinal number), the leftadjoint only for sets with cardinality less than r. In this paper it will be purely categorically shown what from the point of view of universal algebra is wellknown, that the remaining free algebras over a given r-ary theory can be obtained as injective limits of free algebras over sets with cardinality less than r. The proof stems from the following two observations: first that all sets are injective limits of the r-directed system of their subsets with cardinality less than r and secondly that for all sets m with cardinality less than r the Hom-functor <m,> preserves injective limits of r-directed systems of sets. In an appendix the endofunctors of the category of sets which preserve injective limits of r-directed systems of sets are characterized by the same property that the functor-part of a triple in the sets has by definition, if r is the rank of the triple [4].