# Lattice

[https://en.wikipedia.org/wiki/Lattice\_(order](https://en.wikipedia.org/wiki/Lattice_\(order))

A lattice is an abstract structure consisting of a **poset** in which every two elements have a unique supremum (least upper bound or **join**) and a unique infimum (greatest lower bound or **meet**). An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor.

A lattice is an abstract structure studied in the mathematical subdisciplines of _order theory_ and _abstract algebra_.

A **lattice** consists of a _partially ordered set_ in which every two elements have a unique **supremum** (**least upper bound** or **join**) and a unique **infimum** (**greatest lower bound** or **meet**).

An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple (LCM) and the unique infimum is the greatest common divisor (GCD).

1 2 2 1 2 4 2 2 3 4 4 2 2 6 1 2 2 4 6 ↓ ↓ ↓ ↓ ↓ ↓

| 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 | D  |
| -------------------------------------------------------------------------- | -- |
| 1 1 1 1 1 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 | 1  |
| 2   2   2    2     2     2  3  2     2     2     2     2     2     2     2 | 2  |
| 3     3        3                 3        3        3        3        3     | 3  |
| 4       4          4           4           4           4           4       | 4  |
| 5          5              5              5              5           ⅙  5   | 5  |
| 6              6                 6                 6              ⅕  6     | 6  |
| 7                  7                    7                ¼   7             | 7  |
| ⅟   8             ½        8                  ⅓    8                       | 8  |
| 9                          9                          9                    | 9  |
| 10                            10                            10             | 10 |
| 11                               11                                        | 11 |
| 12                                  12                                     | 12 |
| 13                                     13                                  | 13 |
| 14                                        14                               | 14 |
| 15                                           15                            | 15 |
| 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30                               | n  |
