Define a second order tree to be a map between trees (with fixed codomain). We show that many properties of ordinary trees have analogs for second order trees. In particular, we show that there is a notion of “definition by recursion on a well-founded second order tree” which generalizes “definition by transfinite recursion”. We then use this new notion of definition by recursion to prove an analog of Lusin’s Separation theorem for closure spaces of global sections of a second order tree.

Let ${{M}_{n}}$ be the algebra of all $n\,\times \,n$ matrices over $\mathbb{C}$. We say that $A,B\in {{M}_{n}}$ quasi-commute if there exists a nonzero $\xi \,\in \,\mathbb{C}$ such that $AB\,=\,\xi BA$. In the paper we classify bijective not necessarily linear maps $\Phi :{{M}_{n}}\to {{M}_{n}}$ which preserve quasi-commutativity in both directions.

We introduce the notion of strongly projective graph, and characterise these graphs in terms of their neighbourhood poset. We describe certain exponential graphs associated to complete graphs and odd cycles. We extend and generalise a result of Greenwell and Lovász [6]: if a connected graph $G$ does not admit a homomorphism to $K$, where $K$ is an odd cycle or a complete graph on at least 3 vertices, then the graph $G\,\times \,{{K}^{S}}$ admits, up to automorphisms of $K$, exactly $s$ homomorphisms to $K$.