Define a relational structure (A,R) to consist of a set A and a n-ary relation on A for some ordinal n. A homomorphism f:(A,R)->(B,S) is a function f:A->B between the underlying sets of two relational structures of the same arity, such that f(R) c S. Write Str_n for the category of all n-ary relational structures and homomorphisms between them, and Str for the sum in CAT of Str_n over all ordinals n. 1. What is the term for the notion of full subcategory of either Str_n or Str? Failing that, what would be a suitable term? Relational categories? Categories of relational structures? Familiar examples of such categories and an upper bound on their arities include those of groups (3), rings (4), fields (4), lattices (3), vector spaces over the field K (1+max(|K|,2)), directed graphs (2), posets (2), and categories (3). I would expect complete lattices, complete Boolean algebras, topological spaces, and hypergraphs not to embed in Str_n for any n, because their structural elements (subsets, e.g. open sets) have no fixed cardinality bound, unlike n-tuples. A pointer to a proof of such nonembedding, or a demonstration of how to embed them, would be greatly appreciated. Since the concept is such a natural one, this question must surely have been looked at before. 2. What generalizations of Str or Str_n have been considered? One can make Str a bit more "communicative" by permitting homomorphisms between dissimilar structures (A,R), (B,S) of respective arities m<n. Do this by padding out (A,R) to arity n by defining RÂ(n) to be the set of n-tuples of A whose first m elements satisfy R. This probably looks about as useful today as the telephone did in 1879.