0. A finite version

Theorem. (Foster) Let $A$ be a finite algebra such that Var$(A)$ is congruence-distributive. Let $B \in$   Var$(A)$ be finite and subdirectly irreducible. Then $B \in$   HS$(A)$.

Corollary. Under the same hypotheses, $\vert B\vert \leq \vert A\vert$, and if $\vert B\vert = \vert A\vert$ then $B \cong A$.

Example. Each of the lattices $M _ 3$, $N _ 5$ satisfies a law that fails in the other.

Proof of the theorem: Var$(A) =$   HSP$(A)$, so represent $B$ as a homomorphic image of a subalgebra $C$ of $A \times \dots \times A$: $C \subseteq A \times \dots \times A$ and $\phi: C \to B$ (a surjection). Here we know a finite product will do since $B$ is the image of a free algebra Var$_A(n)$, where $n = \vert B\vert$, and such a free algebra can be constructed by the table method. See the left-hand side of Figure .

Figure: Mappings for Foster's Theorem

Focus on Con$(C)$. One of its elements is $\ker \phi$, which by the Correspondence Theorem is meet-irreducible. Some other elements are the kernels of the coordinate projections restricted to $C$: $\ker (\pi_i\vert _ C)$. Of course $\pi_i\vert _ C$ may not map $C$ onto $A$; its image is some subalgebra $S_i$ of $A$.

Observe that

$$\cap _ i \ker (\pi_i\vert _ C) = 0 \leq \ker \phi.$$

Recall that in a distributive lattice, a meet-irreducible element is meet-prime. Therefore $\ker (\pi_{i_0}\vert _ C) \leq \ker \phi$ for some $i_0$. This says that $\pi_{i_0}(a) = \pi_{i_0}(a') \Rightarrow \phi(a) = \phi(a')$. Therefore a well defined map $\psi$ of the image of $S_{i_0}$ onto $B$ is obtained by setting $\psi(\pi_{i_0}(a)) = \phi(a)$. This map is the desired homomorphism showing that $B \in$   HS$(A)$. See the right-hand side of Figure . $\square$