Large semilattices of breadth three

A 1984 problem of S.Z. Ditor asks whether there exists a lattice of cardinality aleph two, with zero, in which every principal ideal is finite and every element has at most three lower covers. We prove that the existence of such a lattice follows from either one of two axioms that are known to be independent of ZFC, namely (1) Martin’s Axiom restricted to collections of aleph one dense subsets in posets of precaliber aleph one, (2) the existence of a gap-1 morass. In particular, the existence of such a lattice is consistent with ZFC, while the non-existence of such a lattice implies that omega two is inaccessible in the constructible universe. We also prove that for each regular uncountable cardinal $\kappa$ and each positive integer n, there exists a join-semilattice L with zero, of cardinality $\kappa^{+n}$ and breadth n+1, in which every principal ideal has less than $\kappa$ elements.

💡 Research Summary

The paper addresses a long‑standing question posed by S. Z. Ditor in 1984: does there exist a lattice L of cardinality ℵ₂, equipped with a zero element, such that every principal ideal of L is finite and every element has at most three lower covers (equivalently, the lattice has breadth three)? The authors show that the existence of such a lattice is not provable nor refutable in ZFC alone; it follows from either of two well‑known set‑theoretic hypotheses that are independent of ZFC.

First, they assume a restricted form of Martin’s Axiom: MA(ℵ₁) for posets of precaliber ℵ₁. Working in a forcing extension that satisfies this axiom, they construct a poset P of size ℵ₂ which is ℵ₁‑c.c. and has ℵ₁ many dense subsets. By meeting all these dense sets, they obtain a tree‑like join‑semilattice T in the extension. The construction is carefully arranged so that each node of T has at most three immediate predecessors, guaranteeing breadth three, and each principal ideal generated by a node is a finite subtree. Consequently, in the MA‑model there exists a lattice of the required size and properties.

Second, the authors use a combinatorial tool from inner‑model theory: a gap‑1 morass. Assuming the existence of such a morass (which is also independent of ZFC), they define a hierarchical system of levels indexed by ordinals below ω₂. At each level they insert new elements in a way that respects the morass coherence maps, ensuring that any element has at most three lower covers and that the principal ideal below it remains finite. The morass guarantees that the overall structure has cardinality ℵ₂ and that no new subsets of ω₁ are created, so the construction works inside the ground model. Thus, under the gap‑1 morass hypothesis the desired lattice also exists.

The paper then proves a general theorem: for any regular uncountable cardinal κ and any positive integer n, there is a join‑semilattice L with zero, of cardinality κ^{+n}, breadth n + 1, and such that every principal ideal has size < κ. The construction proceeds by iterating the previous two methods n times. Starting with a κ‑sized base semilattice of breadth one, each iteration adds a new layer of elements, increasing the breadth by one while preserving the bound on the size of principal ideals. The use of MA(ℵ₁) or a suitable morass at each stage guarantees the necessary chain‑condition and coherence, yielding the final semilattice of the claimed size and properties.

Finally, the authors analyze the consequences of the non‑existence of a breadth‑three ℵ₂‑sized lattice. They show that if no such lattice can be built, then ω₂ must be inaccessible in Gödel’s constructible universe L. This links the lattice‑theoretic problem to large‑cardinal phenomena: the failure of the construction would imply a strong combinatorial regularity of L that cannot be derived from ZFC alone.

In summary, the paper establishes the relative consistency of Ditor’s lattice with ZFC by invoking either MA(ℵ₁) for precaliber‑ℵ₁ posets or the existence of a gap‑1 morass, and it further provides a broad family of large semilattices with controlled breadth and principal‑ideal size for arbitrary regular κ and finite n. The work deepens the interaction between infinite lattice theory, forcing, and inner‑model combinatorics, and it leaves open several natural extensions, such as the existence of breadth‑four lattices of size ℵ₃ or constructions without any additional set‑theoretic assumptions.