Lattice

Remarks

The two forms are equivalent. From a lattice order, the join and meet satisfy clauses 1 to 3. From the operations, the order is recovered by $x \le y \iff x \vee y = y$, which holds exactly when $x \wedge y = x$. The idempotent laws $x \vee x = x$ and $x \wedge x = x$ follow from the absorption laws in clause 3. Example: $\N_{>0}$ ordered by divisibility is a lattice with $a \vee b = \lcm(a, b)$ and $a \wedge b = \gcd(a, b)$. Indeed $\lcm(a, b)$ is a common multiple of $a$ and $b$ and divides every common multiple. Non-example: the poset $\set{a, b, c, d}$ with $a, b < c$ and $a, b < d$ and no other strict relations is not a lattice. The elements $a$ and $b$ have the two upper bounds $c$ and $d$, which are incomparable, so $a \vee b$ does not exist.