This reduction operator (reducer) merges nonextremal elements with the same File number and reduces max in RANK by 1. Thus, using the lattice-coordinate, [Rank #, File #], WE HAVE:

                                 DISTRIBUTIVE
                                     [3,1] max
                                      /\
                                     /  \
                                    /  | \
                                   /   |  \
                                  /    |   \
                                 /     |    \
                                /      |     \
                               /       |      \
                              /        |       \
                             /         |        \
                           [2,1]     [2,2]     [2,3]
                            |\        / \       /|
                            | \      /   \     / |
                            |  \    /     \   /  |
                            |   \  /       \ /   |
                            |     /         \    |
                            |    / \       / \   |
                            |   /   \     /   \  |
                            |  /     \   /     \ |
                           [1,1]     [1,2]     [1,3]
                             \         |       /
                              \        |      /
                               \       |     /
                                \      |    /
                                 \     |   / 
                                  \    |  /
                                   \   | /
                                     [0,1] min

By reduction -- given [X] º [[1,1],[2,1]], [Y] º [[1,2],[2,2]], [Z] º [[1,3],[2,3]] -- this becomes:

                                        MODULAR                  
                                         [3,1] max                       
                                          /|\                       
                                         / | \                     
                                        /  |  \                   
                                       /   |   \                 
                                     [X]  [Y]  [Z]               
                                       \   |   /               
                                        \  |  /                 
                                         \ | /                   
                                          \|/                     
                                          [0,1] min

By the subjector operation, one of modular element of Rank 1 is subjected to another one, raising the latter and max in Rank, achieving a nomodular lattice, as follows:

                                             NONMODULAR
                                               [3,1]
                                                / \
                                               /   \
                                              /    Z=[2,1] 
                                             /      |
                                            /       |
                                          X=[1,1]  Y=[1,2]
                                            \      /
                                             \    /
                                              \  /
                                               \/
                                              [0.1] min