Sublattice

A subset $A$ of a lattice that is closed under the operations $+$ and $\cdot$, i.e. a subset $A$ such that $a+b \in A$ and $ab \in A$ for any $a$ and $b$ from $A$. Therefore, a sublattice is a subalgebra of the lattice considered as a universal algebra with two binary operations. A sublattice $A$ is called convex if $a,b \in A$ and $a\leq c\leq b$ imply $c\in A$. An example of a sublattice is any one-element subset of a lattice; other examples are: an ideal, a filter and an interval. All these sublattices are convex. Any subset in a chain is a sublattice of it (not necessarily convex). The sublattices of a given lattice, ordered by inclusion, form a lattice.

References

[1]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[2]	L.A. Skornyakov, "Elements of lattice theory" , A. Hilger & Hindushtan Publ. Comp. (1977) (Translated from Russian)
[3]	G.I. Zhitomirskii, "Lattices of subsets" , Ordered sets and lattices , 7 , Saratov (1983) pp. 69–97 (In Russian)
[4]	G. Grätzer, "General lattice theory" , Birkhäuser (1978) (Original: Lattice theory. First concepts and distributive lattices. Freeman, 1978)