[Paper Review] A non-coordinatizable sectionally complemented modular lattice with a large J\\'onsson four-frame

Friedrich Wehrung|arXiv (Cornell University)|Mar 25, 2010

Rings, Modules, and Algebras25 references4 citations

TL;DR

This paper constructs a non-coordinatizable sectionally complemented modular lattice of cardinality ℵ₁ with a large 4-frame, resolving an open problem posed by Jónsson in 1962. Using Banaschewski functions and unit-regular rings of index 3, it demonstrates that the existence of a large 4-frame does not imply coordinatizability, even when the lattice is an ideal in a coordinatizable complemented modular lattice with a spanning 5-frame.

ABSTRACT

A sectionally complemented modular lattice L is coordinatizable if it is isomorphic to the lattice L(R) of all principal right ideals of some von Neumann regular (not necessarily unital) ring R. We say that L has a large 4-frame if it has a homogeneous sequence (a_0,a_1,a_2,a_3) such that the neutral ideal generated by a_0 is L. J\\'onsson proved in 1962 that if L has a countable cofinal sequence and a large 4-frame, then it is coordinatizable; whether the cofinal sequence assumption could be dispensed with was left open. We solve this problem by finding a non-coordinatizable sectionally complemented modular lattice L with a large 4-frame; it has cardinality aleph one. Furthermore, L is an ideal in a (necessarily coordinatizable) complemented modular lattice with a spanning 5-frame. Our proof uses Banaschewski functions. A Banaschewski function on a bounded lattice L is an antitone self-map of L that picks a complement for each element of L. In an earlier paper, we proved that every countable complemented modular lattice has a Banaschewski function. We prove that there exists a unit-regular ring R of cardinality aleph one and index of nilpotence 3 such that L(R) has no Banaschewski function.

Motivation & Objective

• To resolve a longstanding open question in lattice theory: whether a sectionally complemented modular lattice with a large 4-frame must be coordinatizable.
• To construct a counterexample to Jónsson’s 1962 conjecture that a large 4-frame implies coordinatizability without requiring a countable cofinal sequence.
• To demonstrate that ideals in coordinatizable complemented modular lattices need not be coordinatizable, even with a spanning 5-frame.
• To establish the non-first-order definability of the class of coordinatizable sectionally complemented modular lattices with a large 4-frame.

Proposed method

• Constructing a unit-regular ring R of cardinality ℵ₁ and index of nilpotence 3 such that L(R), the lattice of principal right ideals, admits no Banaschewski function.
• Using the Condensate Lifting Lemma to lift a Banaschewski function from a quotient lattice to a larger lattice, enabling the construction of a non-coordinatizable lattice with a large 4-frame.
• Employing direct limits and finite products of lattices to build a complemented modular lattice L′ with a spanning 5-frame, ensuring L′ is coordinatizable.
• Proving that the constructed lattice L is isomorphic to an ideal in L′, leveraging the structure of ideals in products of rings and the behavior of homogeneous sequences.
• Applying the theory of Banaschewski traces to show that the existence of a Banaschewski trace is equivalent to coordinatizability in lattices with a large 4-frame.
• Using model-theoretic techniques, including elementary sublattices and first-order logic, to prove that the class of coordinatizable lattices with a large 4-frame is not first-order definable.

Experimental results

Research questions

• RQ1 Does the existence of a large 4-frame in a sectionally complemented modular lattice imply coordinatizability, even without a countable cofinal sequence?
• RQ2 Can a non-coordinatizable sectionally complemented modular lattice with a large 4-frame exist, and if so, what structural properties must it satisfy?
• RQ3 Is the class of coordinatizable sectionally complemented modular lattices with a large 4-frame first-order axiomatizable?
• RQ4 Can an ideal in a coordinatizable complemented modular lattice fail to be coordinatizable, even when the ambient lattice has a spanning 5-frame?
• RQ5 Does the absence of a Banaschewski function in a lattice imply non-coordinatizability, particularly in the context of unit-regular rings?

Key findings

• A non-coordinatizable sectionally complemented modular lattice L of cardinality ℵ₁ with a large 4-frame is explicitly constructed, providing a negative answer to Jónsson’s 1962 question.
• The lattice L is isomorphic to an ideal in a complemented modular lattice L′ with a spanning 5-frame, and L′ is coordinatizable, showing that ideals in coordinatizable lattices need not be coordinatizable.
• A unit-regular ring R of cardinality ℵ₁ and index of nilpotence 3 is constructed such that L(R) has no Banaschewski function, which implies L(R) is not coordinatizable.
• The lattice L is 4/5-entire, and every principal ideal of L is coordinatizable, despite L itself not being coordinatizable.
• The class of coordinatizable sectionally complemented modular lattices with a large 4-frame is not first-order definable, as shown by constructing a countable elementary sublattice of L that is coordinatizable while L is not.
• The existence of a Banaschewski trace is shown to be equivalent to coordinatizability in lattices with a large 4-frame, and the constructed lattice L has no such trace, confirming its non-coordinatizability.