Cardinality of pairwise non-isomorphic complete lattices on an infinite cardinal $\kappa$

Question

Suppose $\kappa$ is an infinite cardinal. Let $\cal S$ be a collection of pairwise non-isomorphic complete lattices on the ground set $\kappa$. What cardinality can $\cal S$ have at most? Is the answer different if we restrict ourselves to complete distributive lattices?

Answer 1

There is a trivial upper bound of $2^\kappa$. In fact, this bound can already be achieved for complete totally ordered sets. For instance, given any subset $A\subseteq\kappa+1$, let $L_A$ be obtained from $\kappa+1$ by replacing each element of $A$ with a copy of $\mathbb{Z}\cup\{-\infty,\infty\}$. Then $L_A\cong L_B$ iff $A=B$, since the obvious surjection $L_A\to\kappa+1$ can be canonically defined by collapsing every closed interval in $L_A$ which is isomorphic to $\mathbb{Z}\cup\{-\infty,\infty\}$ to a point, and $\kappa+1$ is rigid.