Lattices and semilattices derived from commutative rings of characteristic 2 satisfying the identity $x^{2^n}\approx x$

We prove that a commutative ring $\mathbf R=(R,+,\cdot)$ of characteristic $2$ satisfying the identity $x^{2^n}\approx x$ together with the binary relation $\le$ on $R$ defined by $x\le y$ if $xy=x^2$ forms a meet-semilattice with smallest element $0$.

If, moreover, $\mathbf R$ is unitary then we derive two binary term operations $\wedge$ and $\vee$ on $R$ which together with the unary term operation $x':=x+1$ form a Boolean algebra.