An example of the decomposition non-uniqueness for a 3-dimensional geometric object

In 1942, M.H.A. Newman formulated and proved a simple lemma that has been very useful in various areas of mathematics in particular in algebra and Gröbner — Shirshov bases theory. It was later called Diamond Lemma, since its key design is graphically depicted as a rhombus (diamond symbol). In 2005, I proposed a new version of this lemma, designed to solve geometric problems, and proved existence and uniqueness theorems for primary decompositions of various geometric objects: 3-dimensional manifolds, knots in thickened surfaces, knotted graphs, knotted theta curves in 3-dimensional manifolds. It turned out that all geometric objects of the mentioned types allow primary decomposition, but in some cases (for example, for orbifolds) uniqueness decomposition is absent. This article presents this new version of the lemma and an algorithm for its application. I propose a theorem that uses Diamond Lemma to prove it, and a counterexample showing the impossibility of omitting one of the conditions of the theorem.