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arXiv · 2608.08723

Approximate homomorphisms on orthomodular lattices

Tomasz Kania
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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.

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Tomasz Kania. 2026-08-09. Approximate homomorphisms on orthomodular lattices. https://arxiv.org/abs/2608.08723

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