On the Finiteness Problem for classes of modular lattices

TL;DR

This paper proves that determining whether a large class of modular lattices is finite is an unsolvable problem, highlighting fundamental limitations in classifying such algebraic structures.

Contribution

It establishes the undecidability of the Finiteness Problem for broad classes of modular lattices, extending understanding of computational limits in lattice theory.

Findings

Finiteness Problem is undecidable for large classes of modular lattices

Shows fundamental computational limitations in classifying modular lattices

Highlights the boundary of algorithmic solvability in algebraic structures

Abstract

The Finiteness Problem is shown to be unsolvable for any sufficiently large class of modular lattices.

Equations

$$y_i\cdot y_j=y_{\perp }(1\le i<j\le 4),y_i+y_j=y_{\top }(1\le i<j\le 4,j\ne 2)$$

Full text

Given a class $\mathcal{A}$ of algebraic structures, the Finiteness Problem is to decide for any given finite presentation, that is a list of generator symbols and relations, whether or not there is a finite bound on the size of members of the class which ’admit the presentation’, that is a system of generators satisfying the given relations; if $\mathcal{A}$ is a quasi-variety, this means finiteness of the free $\mathcal{A}$-algebra given by the presentation. Due to Slavik [6], the finiteness problem is algorithmically solvable for the class of all lattices, due to Wille [7] for any class of modular lattices, containing the subspace lattice of an infinite projective plane, if one allows only order relations between the generators. The present note relies on the unsolvability of the Triviality Problem for modular lattices [4] which in turn relies on the result of Adyan [1, 2] and Rabin [5] for groups. For a vector space $V$ let $\operatorname{L}(V)$ denote the lattice of subspaces.

The following restates the relevant part of Lemma 10 in [4].

1. (i)

If, for $\varphi\in\Sigma$, $\varphi^{\exists}$ is valid in some modular lattice, then it is so within $\operatorname{L}(V)$ for any $V$ of infinite dimension. Moreover, one can choose $x_{\bot}=0$ and $x_{\top}=V$. 2. (ii)

The set of all $\varphi\in\Sigma$ with $\varphi^{\exists}$ valid in some modular lattice is not recursive.

Consider the conjunction $\pi(\bar{y},y_{\bot},y_{\top})$ of the following lattice equations

[TABLE]

We use $x,y,\ldots$ both as variables and generator symbols and also to denote their values under a particular assignment. In [3], $\operatorname{FM}(J_{4}^{1})$ was defined as the modular lattice freely generated under the presentation $\pi(\bar{y},y_{\bot},y_{\top})$ (equivalently, by the partial lattice $J^{4}_{1}$ arising from the $6$-element height $2$ lattice $M_{4}$ with atoms $y_{1},y_{2},y_{3},y_{4}$ keeping all joins and meets except the join of $\{y_{1},y_{2}\}$). The following was shown (to prove (i) consider $V$ the direct sum of infinitely many subspaces of dimension $\aleph_{0}$).

1. (i)

$\operatorname{FM}(J_{4}^{1})$* embeds into $\operatorname{L}(V)$ for any $V$ of infinite dimension. Moreover, the embedding can be chosen such that any prime quotient has infinite index.* 2. (ii)

$\operatorname{FM}(J_{4}^{1})$* has infinite height.* 3. (iii)

$\operatorname{FM}(J_{4}^{1})$* has prime quotient $y_{\top}/(y_{1}+y_{2})$, generating the unique proper congruence relation $\theta$.* 4. (iv)

$\operatorname{FM}(J_{4}^{1})/\theta$* is isomorphic to $M_{4}$.*

1. (i)

$x_{\bot}=x_{\top}$. 2. (ii)

$L$ is singleton or $M_{4}$. 3. (iii)

$L$ is finite. 4. (iv)

$L$ is of finite height.

Clearly, if $x_{\bot}=x_{\top}$ in every modular lattice admitting presentation $\varphi$ then the same applies to the presentation $\varphi^{\#}$. On the other hand, assume that $\varphi^{\exists}$ is valid in some modular lattice. Given any vector space $V$, embed $\operatorname{FM}(J_{4}^{1})$ into $\operatorname{L}(V)$ as in (i) of Lemma 3 and denote $U=y_{1}+y_{2}$. By (i) of Lemma 2 one can evaluate $\bar{x}$ in $\operatorname{L}(V/U)$ such that $\varphi(\bar{x},x_{\bot},x_{\top})$ holds where $x_{\bot}=U$ and $x_{\top}=V$. This results into generators of a sublattice $L$ of $\operatorname{L}(V)$ satisfying the relations of $\varphi^{\#}$ and such that $x_{\bot}\neq x_{\top}$. Thus, to decide whether $x_{\bot}=x_{\top}$ for all modular lattices admitting presentation $\varphi$ reduces to deciding whether (i)–(iv) apply to all $L\in\mathcal{A}$ admitting presentation $\varphi^{\#}$. Undecidability of the latter problems follows now from (ii) of Lemma 2. ∎