Category:Sublattices

This category contains results about Sublattices.
Definitions specific to this category can be found in Definitions/Sublattices.

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 $\struct {L, \wedge_L, \vee_L, \preceq_L}$

if and only if:

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