We know that $(L, \vee, \wedge)$ is called a lattice if $(L, \vee)$ is a join-semilattice and $(L, \wedge)$ is a meet-semilattice, i.e $(L, \vee)$ and $(L, \wedge)$ are idempotent commutative semigroups. However, the Wikipedia article gives another definition for lattice additionally: $(L, \vee, \wedge)$ is called a lattice if $(L, \vee)$ and $(L, \wedge)$ are commutative semigroups and the absorption law holds:
$$\forall a, b, c \in L:\;\; a \vee (a \wedge b) = a \;\;\;\;\;\;and\;\;\;\;\;\; a \wedge(a \vee b) = a$$
Now, the second definition implies the first one but is the converse true? I think it's not and actually the second definition is stronger.
Are these two definitions of lattice equivalent
lattice-orders
Best Answer
The second definition does not follow from the first, as witnessed by the lattice $(\{0, 1\}, \max, \max)$.