Namespaces
Variants
Actions

Difference between revisions of "Centre of a partially ordered set"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
−
The set of elements of a partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021260/c0212601.png" /> with a 0 and a 1 (in particular, of a lattice) for which in some decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021260/c0212602.png" /> into a direct product one of the components is 1 and the other is 0. The centre of any partially ordered set with a 0 and a 1 is a Boolean algebra. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021260/c0212603.png" /> of a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021260/c0212604.png" /> belongs to the centre if and only if it is neutral (that is, if every triple of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021260/c0212605.png" /> generates a distributive sublattice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021260/c0212606.png" />) and has a complement. In a complemented modular lattice the centre coincides with the set of all elements having a unique complement.
+
{{TEX|done}}
 +
The set of elements of a partially ordered set $P$ with a 0 and a 1 (in particular, of a lattice) for which in some decomposition of $P$ into a direct product one of the components is 1 and the other is 0. The centre of any partially ordered set with a 0 and a 1 is a Boolean algebra. An element $a$ of a lattice $L$ belongs to the centre if and only if it is neutral (that is, if every triple of elements $\{a,x,y\}$ generates a distributive sublattice of $L$) and has a complement. In a complemented modular lattice the centre coincides with the set of all elements having a unique complement.

Latest revision as of 13:40, 31 July 2014

The set of elements of a partially ordered set $P$ with a 0 and a 1 (in particular, of a lattice) for which in some decomposition of $P$ into a direct product one of the components is 1 and the other is 0. The centre of any partially ordered set with a 0 and a 1 is a Boolean algebra. An element $a$ of a lattice $L$ belongs to the centre if and only if it is neutral (that is, if every triple of elements $\{a,x,y\}$ generates a distributive sublattice of $L$) and has a complement. In a complemented modular lattice the centre coincides with the set of all elements having a unique complement.

How to Cite This Entry:
Centre of a partially ordered set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Centre_of_a_partially_ordered_set&oldid=12802
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Centre_of_a_partially_ordered_set&oldid=32614"