Post algebra

Remarks

Example: $\mathbf{P}_n^2$ is a Post algebra of order $n$, since a variety is closed under direct products. Non-example: the three-element chain $(\set{0, 1, 2}, \max, \min, ', 0, 2)$ with $x' = 2 - x$ is not a Post algebra of order $3$. In $\mathbf{P}_3$ the identity $x''' \approx x$ holds, but here $0''' = 2 \ne 0$. The algebra $\mathbf{P}_n$ is primal, because $\max$ and the cyclic negation already generate every operation on $P_n$, a result of Post. For $n = 2$, $\mathbf{P}_2$ is the two-element Boolean algebra, so Post algebras of order $2$ are Boolean algebras. Other texts present Post algebras through lattice axioms, as bounded distributive lattices with a chain of constants $e_0 < e_1 < \cdots < e_{n - 1}$. Post algebras give algebraic semantics for Post's many-valued logic.