A 0-simple lattice which is not a 0,1-simple lattice.

Question

A 0,1-simple lattice is a non-trivial bounded lattice $L$ such that non-constant homomorphisms of $L$ preserve the identity of its top and bottom elements. I got this definition from Wikipedia. I also have made up my own definition of a 0-simple lattice. I define a 0-simple lattice to be a non-trivial bounded lattice $L$ such that non-constant homomorphism of $L$ preserve the identity of its bottom element. In other words, it only satisfies half of the definition of a 0,1-simple lattice. Is there an example of a 0-simple lattice which is not a 0,1-simple lattice?

Answer 1

Here is a construction of one. Start with the simple lattice $M_3$. It has least element $0$, largest element $1$, and three incomparable middle elements $a, b, c$. Now add three new elements $a', c', 1'$ just above $a, c, 1$. That is: $1'$ will be a new top element, $a'$ is a new element above $a$ but not above $1$, $c'$ is a new element above $c$ but not above $1$. I will add a sketch.

The only congruences of the lattice I sketched are the equality relation $\Delta$, the universal relation $\nabla$, and the congruence $\theta$ whose associated partition is $\{\{0\},\{a,a'\},\{b\},\{c,c'\},\{1,1'\}\}$. To see that this lattice is $0$-simple, one must show that any proper congruence relates no non-$0$ element to $0$. (Neither $\Delta$ nor $\theta$ does this.) To see that it is not $1'$-simple, one should exhibit a proper congruence that relates some non-$1'$ element to $1'$. (Use $\theta$.)