Subtiles and supertiles

Some times is useful to have this definitions in order to obtain tiles from others tiles, or for reduce the tile's appearance and preserve the validity of the last definitions.

Definition 5   If $M_1$ and $M_2$ are tiles in $lca(2,1)_{110}$ , we can say that $M_1$ is a subtile of $M_2$ , typed by $M_1\subset M_2$ , when after the removal $M_2$ of $M_1$ , $M_1$ still preserves its tile's properties.

Definition 6   If $M1\subset M_2$ , then we say that $M_2$ is a supertile for $M_1$ , typed by $M_2\supset M_1$ .

Above definitions mean that we can never have a tile after removing triangles (or subtiles), like we can see in figure 5. Otherwise, in figure 6 shows an example in which after removing a tile in another relative position, the tile's conditions are not preserved.

Figure 5: Tile $a$ is composite by copies of the same tile, in $b$ , and the patch that result after removing a tile, preserves the tile's conditions, so we have tiles and super tiles.

Figure 6: After removing a tile from other bigger tile in $a$ , we have a patch that preserve not tile's conditions, then we have neither supertiles nor subtiles.