  We generalize the Brin-Higman-Thompson groups $n G_{k,1}$ to monoids $n
M_{k,1}$, for $n \ge 1$ and $k \ge 2$, by replacing bijections by partial
functions. The monoid $n M_{k,1}$ has $n G_{k,1}$ as its group of units, and is
congruence-simple. Moreover, $n M_{k,1}$ is finitely generated, and for $n \ge
2$ its word problem is {\sf coNP}-complete. We also present new results about
higher-dimensional joinless codes.
