On the equational theory of finite modular lattices

TL;DR

This work proves that there exists a finite bound $N$ for which the $N$-variable equational theory of finite modular lattices is undecidable, by reducing the restricted word problem for finite groups to lattice identities. The core method, Freese's frame technique, encodes finitely presented groups into projective modular lattices via towers of frames and glueing constructions, using stable elements and coordinate rings to force group relations inside lattices. The paper develops a detailed framework of presentations, reductions, and glueing to realize uniform, projective constructions (including towers of 2-, 3-, and skew-frames) that simulate group-theoretic computations within modular lattices. The results extend undecidability to broad classes of finite modular lattices and provide a systematic approach for translating quasi-identities into lattice identities, with potential implications for the decidability of lattice-based theories in algebraic structures.

Abstract

It is shown that there is $N$ such that there is no algorithm to decide for identities in at most $N$ variables validity in the class of finite modular lattices. This is based on Slobodskoi's result that the Restricted Word Problem is unsolvable for the class of finite groups and relies on Freese's technique of capturing group presentations within free modular lattices.

Figures (8)

• Figure 1: Direct product of chains
• Figure 2: $2$-frame and $3$-frame
• Figure 3: Modular lattice generated by $a,a',c$ with $aa'\leq c \leq a+a'$
• Figure 4: Reduction $2$-frame $(\Phi,a_i,c_{12})$ to $(\Phi^b_d, a'_i,c'_{12})$
• Figure 5: Reduction of $\Delta(2)$

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.4
• proof
• Lemma 3.5
• Lemma 3.6
• proof
• Lemma 3.7
• proof
• Corollary 3.8