Algebraic aspects of an ordered band arising in nonlinear signal processing (Q2481320)

Ordered semigroups arise quite naturally in nonlinear signal processing where idempotent monotone operators play an essential role. For any natural numbers \(M\) and \(N\) the author defines a set \({\mathcal P}(M,N)\) of finite sequences of symbols \(L_1,\dots, L_M, U_1,\dots, U_N\) of cardinality \(\left( \begin{smallmatrix} M + N + 2 \\ N + 1 \\ \end{smallmatrix} \right) - 2.\) For \(M=1, N=2\), for example, \[ {\mathcal P}(1, 2)=\{L_1, U_1, U_2, L_1U_1, L_1U_2, U_1L_1, U_2L_1, U_2L_1U_1\}; \] for \(M=2, N=1\), \[ {\mathcal P}(2, 1)=\{L_1, L_2, U_1, L_1U_1, L_2U_1, U_1L_1, U_1L_2, L_2U_1L_1\} \] (for alternating symbols \(L_i\) and \(U_j\), reading from left to right, the indices within the \(L\)-subsequence and the indices within the \(U\)-subsequence are strictly decreasing). He endows the set \({\mathcal P}(M,N)\) with an order relation \(\leq\) and poses the question whether the ordered set \({\mathcal P}(M,N)\) is a lattice. Then he defines a finite band \({\mathcal B}(M,N)\) generated by some symbols \(L_1,\dots, L_M, U_1,\dots, U_M\) and subject to some relations so that \(| {\mathcal B}(M,N)| =| {\mathcal P}(M,N)| \). Finally, the author defines a self-dually ordered band \({\mathcal B{'}}(M,N)\) of monotone operators which is isomorphic as a semigroup to \({\mathcal B}(M,N)\) and isomorphic as an ordered set to \({\mathcal P}(M,N)\) and poses the question whether it is a lattice.