Definition:Sublattice

Definition
Let $\struct {L, \wedge_L, \vee_L, \preceq_L}$ be a lattice.

Let $S$ be a subset of $L$.

Let $\wedge_S$ and $\vee_S$ be the restrictions to $S$ of $\wedge_L$ and $\vee_L$ respectively.

Let $\preceq_S$ be the restriction to $S$ of $\preceq_L$.

Then:
 * $\struct {S, \wedge_S, \vee_S, \preceq_S}$ is a sublattice of $\struct {L, \wedge_L, \vee_L, \preceq_L}$


 * $S$ is closed under $\wedge_S$ and $\vee_S$.
 * $S$ is closed under $\wedge_S$ and $\vee_S$.

If in addition $L$ is a bounded lattice and its top and bottom elements are in $S$, then $S$ is called a $0, 1$-sublattice of $L$.

If in addition $L$ is a complete lattice and for each subset $T$ of $S$, $\map {\sup_L} T, \map {\inf_L} T \in S$, then $S$ is called a complete sublattice of $L$.

Also see

 * Sublattice is Lattice