For cardinals $\kappa,\lambda,\mu$ we let $\mathfrak{b}_{\kappa,\lambda,\mu}$ be the smallest size of a subset B of $^\lambda\mu$ unbounded in the sense of $\leq_\kappa$; that is, such that there is no function $f\in{}^\lambda\mu$ such that $\{\alpha<\lambda:g(\alpha)>f(\alpha)\}$ has size less than $\kappa$ for all $g\in B$. Similarly for $\mathfrak{d}_{\kappa,\lambda,\mu}$, the general dominating number, which is the smallest size of a subset B of $^\lambda\mu$ such that for every $g\in{}^\lambda\mu$ there is an $f\in B$ such that the above set has size less than $\kappa$. These cardinals are generalizations of the usual ones for $\kappa=\lambda=\mu=\omega$. When all three are the same regular cardinal, the relationships between them have been completely described by Cummings and Shelah. We also consider some variants of the functions, following van Douwen, in particular the version $\mathfrak{b}^{\uparrow}_{\kappa,\lambda,\mu}$ of $\mathfrak{b}_{\kappa,\lambda,\mu}$ in which B is required to consist of strictly increasing functions. Some of the main results of this paper are: (1) $\mathfrak{b}_{\mu,\mu,{\rm cf}\mu}\leq\mathfrak{b}_{{\rm cf}\mu,{\rm cf}\mu,{\rm cf}\mu}$; (2) for $\lambda\leq\mu$, $\mathfrak{b}^{\uparrow}_{\kappa,\lambda,\mu}$ always exists; (3) if $\mathrm{cf}\lambda= \mathrm{cf}\mu<\lambda\leq\mu$, then $\mathfrak{b}_{{\rm cf}\mu,{\rm cf}\mu,{\rm cf}\mu}= \mathfrak{b}^{\uparrow}_{\lambda,\lambda,\mu}$; (4) $\mathfrak{d}_{\omega,\mu,\mu}=\mathfrak{d}_{1,\mu,\mu}$. For background see Section 1 of the paper. Several open problems are stated.

We prove that there is no maximal low d.c.e degree.

A reduct of a first-order structure is another structure on the same set with perhaps fewer definable predicates. We consider reducts of the complex field which are proper (not essentially the whole field) but nontrivial in a sense to be made precise in the paper. Our main result lists seven kinds of reducts. The list is complete in the sense that every reduct is a finite cover of one of these. We also investigate when two items on our list can be the same, in a couple of natural senses.