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A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090910/s0909101.png" /> of a [[Lattice|lattice]] that is closed under the operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090910/s0909102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090910/s0909103.png" />, i.e. a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090910/s0909104.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090910/s0909105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090910/s0909106.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090910/s0909107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090910/s0909108.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090910/s0909109.png" />. Therefore, a sublattice is a subalgebra of the lattice considered as a universal algebra with two binary operations. A sublattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090910/s09091010.png" /> is called convex if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090910/s09091011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090910/s09091012.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090910/s09091013.png" />. 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.
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A subset $A$ of a [[Lattice|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 $A$ 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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Skornyakov,  "Elements of lattice theory" , A. Hilger &amp; Hindushtan Publ. Comp.  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.I. Zhitomirskii,  "Lattices of subsets" , ''Ordered sets and lattices'' , '''7''' , Saratov  (1983)  pp. 69–97  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G. Grätzer,  "General lattice theory" , Birkhäuser  (1978)  (Original: Lattice theory. First concepts and distributive lattices. Freeman, 1978)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Skornyakov,  "Elements of lattice theory" , A. Hilger &amp; Hindushtan Publ. Comp.  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.I. Zhitomirskii,  "Lattices of subsets" , ''Ordered sets and lattices'' , '''7''' , Saratov  (1983)  pp. 69–97  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G. Grätzer,  "General lattice theory" , Birkhäuser  (1978)  (Original: Lattice theory. First concepts and distributive lattices. Freeman, 1978)</TD></TR></table>

Revision as of 16:47, 14 January 2012

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 $A$ 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)
How to Cite This Entry:
Sublattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sublattice&oldid=20292
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article