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 $x^{\mathrm{\prime }}:=x+1$ form a Boolean algebra.