Principal congruences on some lattice-ordered algebras (Q914716)

The authors solve the question in which algebras all congruences are principal (PC). They prove: 1) A distributive lattice L has property (PC) if L is finite and all subsets of J(L) \((=\) the poset of all nonzero join-irreducible elements of L) are convex (resp. \(\ell (J(L))\leq 1).\) 2) A Stone algebra L has property (PC) if L is finite and \(\ell (J(L))\leq 2.\) 3) A de Morgan algebra has property (PC) if L is finite and \(\ell (J(L))\leq 3.\) They also show that the Heyting algebras behave quite differently - to ensure that all congruences are principal a chain condition is necessary and sufficient.