Approximate homomorphisms on orthomodular lattices

Abstract

The stability programme initiated by Ulam asks when approximate solutions to algebraic identities must lie near exact ones. For lattices, this leads to the question of when a map that nearly preserves joins and meets can be approximated by a genuine lattice homomorphism. Badora--Kochanek--Przebieracz developed a neighbourhood-based framework for distributive lattices, centred on a separation (sandwich) lemma that constructs an exact join homomorphism between a join-subhomomorphism and a join-superhomomorphism via an order envelope. We revisit this mechanism and identify the single step at which distributivity is used: a decomposition identity for elements lying below a join. Without distributivity, separation can fail already in the modular lattice $M_3$ and in a small finite orthomodular lattice. On the positive side, we show that separation holds in arbitrary lattices whenever the lower bounding map is isotone. For orthomodular lattices---algebraic models of quantum logic---we develop a blockwise stability theory on Boolean blocks. Approximate identities on compatible pairs yield exact homomorphic selections on each block (for joins, for meets, and for both operations under bi-admissibility). We present several gluing criteria for assembling blockwise selections, and we give a concrete finite example showing that gluing can fail when block overlaps are non-trivial. Finally, in the spirit of Kalton--Roberts, we obtain blockwise approximation results for nearly additive functions on orthomodular lattices by finitely additive measures, with an illustration on finite-dimensional projection lattices.