The lattices of kernel ideals in pseudocomplemented De Morgan algebras (Q523142)

A pseudocomplemented De Morgan algebra is a bounded distributive lattice \(L\) endowed with two unary operations, \(\circ\) and \(\ast\), such that \((L,\circ)\) is a De Morgan algebras and \((L,\ast)\) a distributive p-algebra. Pseudocomplemented De Morgan algebras form an equational class \textbf{pdM}. A kernel ideal \(I\) of \(L\) in \textbf{pdM} is a lattice ideal which is the class of zero for some congruence of \(L\), equivalently, a lattice ideal such that for every \(x \in I\), \(x^{\ast \circ} \in I\). Write \(\mathcal{I}_k(L)\) for the complete distributive lattice of kernel ideals of \(L\). Since it is complete, it is a Heyting algebra. Write \(Z(L)\), for the set of elements \(x \in L\) such that \(x^{\circ} = x^{\ast}\). The authors give necessary and sufficient conditions for kernel ideals being a Boolean and a Stone lattice. More concretely, the lattice \(\mathcal{I}_k(L)\) is Boolean if and only if \(Z(L)\) is finite and for every \(x\), there exists an \(N \geq 0\) such that \((x \vee x^{\ast \circ})^{n (\ast \circ)} = (x \vee x^{\ast \circ})^{N (\ast \circ)}\), for \(n \geq N\); and \(\mathcal{I}_k(L)\) is a Stone lattice if and only if \(Z(L)\) is complete and for every \(x\), \(\displaystyle \bigwedge_{n \geq 0}(x \vee x^{\ast \circ})^{\ast n (\ast \circ)}\) exists. Furthermore, defining the class of De Morgan-Heyting algebras in the obvious way, it is shown that for every \(L\) in this class, the lattice of congruences of \(L\) is isomorphic to \(\mathcal{I}_k(L)\).