\(P(2,2)\): If \(X\) is a set and \(P\) is a property of subsets of \(X\) of 2 character (that is, \(\forall y\subseteq X (P(y)\) iff \(\forall z\subseteq y (|z|\le 2 \rightarrow P(z)))\)) then if every finite subset of \(X\) can be partitioned into two or fewer \(P\)-sets (that is, sets \(z\) such that \(P(z))\) then \(X\) can be partitioned into two or fewer \(P\)-sets.

\(G_{2}\): A graph is 2-colorable if every finite subgraph is 2-colorable. note 117.

If \(m\) is a cardinal number and \({\cal L}\) is a lattice isomorphic to the lattice of subalgebras of some unary universal algebra (a unary universal algebra is one with only unary or nullary operations) and \(\alpha\) is an automorphism of \({\cal L}\) of order 2 (that is, \(\alpha ^{2}\) is the identity) then there is a unary algebra \(\frak A\) with \(|\hbox{domain }\frak A| \ge m\) and an isomorphism \(\rho\) from \({\cal L}\) onto the lattice of subalgebras of \(\frak A^{2}\) with \[\rho (\alpha (x)) = (\rho (x))^{-1} (= \{(s,t) : (t,s)\in \rho (x)\})\] for all \(x\in {\cal L}\).

For any cardinal \(m\), there is a unary algebra \(\frak A = \langle A,F\rangle\) (only unary and nullary operations) and a proper subalgebra \(\frak D\) of \(\frak A^{2}\) such that \(\frak A^{2}\) has exactly four subalgebras, \(\frak A\), \(\emptyset\), \(\frak D\) and \(\frak D^{*}= \{(s,t): (t,s)\in\frak D\}\).