Word problems in Elliott monoids

TL;DR

This paper explores the algorithmic complexity of word problems in Elliott monoids, showing polynomial-time solvability for certain AF algebras and constructing quotients with G"odel incomplete problems.

Contribution

It introduces a natural word problem for Elliott local semigroups, proves polynomial-time solvability for key classes, and constructs quotients with G"odel incomplete word problems.

Findings

Word problem of al M_1 is solvable in polynomial time.

Word problems of al A_{n,k} and al F_{ heta} are solvable in polynomial time.

A quotient of al M_1 with Gf6del incomplete word problem is constructed.

Abstract

Algorithmic issues concerning Elliott local semigroups are seldom considered in the literature, although these combinatorial structures completely classify AF algebras. In general, the addition operation of an Elliott local semigroup is {\it partial}, but for every AF algebra $\mathfrak B$ whose Murray-von Neumann order of projections is a lattice, this operation is uniquely extendible to the addition of an involutive monoid $E(\mathfrak B)$. Let $\mathfrak M_1$ be the Farey AF algebra introduced by the present author in 1988 and rediscovered by F. Boca in 2008. The freeness properties of the involutive monoid $E(\mathfrak M_1)$ yield a natural word problem for every AF algebra $\mathfrak B$ with singly generated $E(\mathfrak B)$, because $\mathfrak B$ is automatically a quotient of $\mathfrak M_1$. Given two formulas $\phi$ and $\psi$ in the language of involutive monoids, the problem asks to decide whether $\phi$ and $\psi$ code the same equivalence of projections of $\mathfrak B$. This mimics the classical definition of the word problem of a group presented by generators and relations. We show that the word problem of $\mathfrak M_1$ is solvable in polynomial time, and so is the word problem of the Behnke-Leptin algebras $\mathcal A_{n,k}$, and of the Effros-Shen algebras $\mathfrak F_{\theta}$, for $\theta\in [0,1]\setminus \mathbb Q$ a real algebraic number, or $\theta = 1/e$. We construct a quotient of $\mathfrak M_1$ having a G\"odel incomplete word problem, and show that no primitive quotient of $\mathfrak M_1$ is G\"odel incomplete.