Word problems in Elliott monoids

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 partial, but for every AF algebra B whose Murray-von Neumann order of projections is a lattice, this operation is uniquely extendible to the addition of an involutive monoid E(B). Let M1 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(M1) yield a natural word problem for every AF algebra B with singly generated E(B), because B is automatically a quotient of M1. Given two formulas φ and ψ in the language of involutive monoids, the problem asks to decide whether φ and ψ code the same equivalence of projections of 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 M1 is solvable in polynomial time, and so is the word problem of the Behnke-Leptin algebras An,k, and of the Effros-Shen algebras Fθ, for θ ∈ [0, 1] \Q a real algebraic number, or θ = 1/e. We construct a quotient of M1 having a Gödel incomplete word problem, and show that no primitive quotient of M1 is Gödel incomplete.