Pudlák and Tůma https://link.springer.com/article/10.1007/BF02482893 proved that every finite lattice can be embedded as a sublattice of the partition lattice of a finite set. Can this be generalized in either of the following ways, or preferably both together?

  1. If the finite lattice $L$ can be represented linearly as partitions of some set, can the embedding into the partition lattice of a finite set $B$ always be chosen to be linear? (Here, "linear" means that the representing partitions commute with each other, in that if $x, y, z \in B$ and $x \equiv y$ in representative $\phi(a)$ and $y \equiv z$ in representative $\phi(b)$, then there exists $y^\prime \in B$, $x \equiv y^\prime$ in $\phi(b)$, and $y^\prime \equiv z$ in $\phi(a)$. This is a strong condition on $L$, including that it is Arguesian.)

  2. Relax the condition on $L$ from finite to finitary, that is, every principal (lower) ideal is finite. Clearly $L$ as a whole cannot be represented by partitions of any finite set $B$. But can we require that given any principal ideal $[\hat{0}, x]$ of $L$, restricting the representation to the ideal makes it "quasi-finite" in some suitable sense? I think the natural requirement is that the representative $\phi(x)$ partitions $B$ into an (infinite) number of finite blocks. (This implies that all blocks of all representatives of elements of $[\hat{0}, x]$ are subsets of one of these finite blocks, so all of the representatives are dissected simultaneously into an infinite set of representative partitions on the finite blocks.) Restricting the representation to one of the finite blocks will not necessarily be a representation of the ideal as the restricted representation may not be one-to-one, but necessarily there is a finite subset of the finite blocks on whose union the effect of the representation will be one-to-one from the ideal to the union of the blocks.

A converse of (1) would be, if every principal ideal of a finitary lattice L is linear, can all of those linear representations be assembled into a linear representation of L? This is a version of the problem of a direct limit of linear lattices: It's known that a direct limit of linear lattices is itself a linear lattice, but there is no known natural way to construct a linear representation of the limit from the linear representations of the sublattices. Haiman, https://www.sciencedirect.com/science/article/pii/0001870885901185, sec. 1.0; Jónsson, https://www.ams.org/journals/tran/1959-092-03/S0002-9947-1959-0108459-5/, sec. 3, Th. 2.



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