Quasi-variety status of the class of quantum models for a fixed signature

Determine whether, for any fixed signature Δ=(Σ,E,Prop) of hybrid-dynamic quantum logic, the class Mod(Δ) of quantum models—Kripke structures (W,M) where W is a first-order model of (Σ,E) whose Hilbert-space reduct W_{Σ^h} is a Hilbert space, each symbol u in the set U of unitary transformations denotes a unitary operator on W_v, each symbol q in the set Q of measurements denotes a projective measurement q^W(w)=P_X(w)/sqrt(pr_X(w)) for some closed subspace X, and for each closed propositional symbol r in Prop^c the set {w in W_v | r in M_w} is a closed subspace—forms a quasi-variety.

Background

The paper develops initial semantics for hybrid-dynamic quantum logic using inductive arguments and explicitly notes that these results avoid the heavier model-theoretic machinery of quasi-varieties and inclusion systems previously employed for hybridized institutions. Establishing whether the relevant class of models is a quasi-variety would connect these results to broader algebraic and institutional frameworks for initial semantics.

Quantum models in this setting are Kripke structures whose states are vectors in a Hilbert space, with actions interpreted as unitary transformations and projective measurements, and propositional labels constrained so that closed propositional symbols denote closed subspaces. Clarifying whether the entire class Mod(Δ) is a quasi-variety would have implications for axiomatizability and closure properties central to general model-theoretic approaches.