On basis elements in idempotent subspaces (Q1277590)

Let \(\Omega\subset \mathbb{R}^n\) be a solid, and let \[ H\Biggl({\partial S\over\partial x}, x\Biggr)= E\tag{1} \] be the Hamilton-Jacobi (Bellman) equation, where \(H(p,z)\in C^\infty\) is a function convex in \(p\) and invariant with respect to a transformation group \(U\): \(UH(p,x)= H(p,x)\). Let \(M\) be the space \((\min,+)\) in \(\Omega\); i.e., the ``addition'' is \(a\oplus b= \min(a,b)\) and the ``multiplication'' is \(a\odot b= a+b\). If \(S_1\) and \(S_2\) are solutions to (1), then \(C_1\odot S_1\oplus C_2\odot S_2\) is also a (generalized) solution. Theorem. Let \(S(x)\) is a basis element in the idempotent subspace of (generalized) solutions to (1). Then \(US(x)\) is also a basis element.