Partial linear space

Remarks

Example: the $3 \times 3$ grid, with $P = \set{1, 2, 3}^2$ and $L$ the three rows $\set{i} \times \set{1, 2, 3}$ and the three columns $\set{1, 2, 3} \times \set{j}$. A row and a column meet in one point, two rows or two columns are disjoint, so two distinct points share at most one line; each line has three points.

Non-example: $P = \set{a, b, c}$ with lines $\set{a, b}$ and $\set{a, b, c}$. The points $a$ and $b$ lie on two lines, so clause 1 fails.

Clause 1 is equivalent to its dual: two distinct lines meet in at most one point. Partial linear spaces are also called semilinear spaces or near-linear spaces.