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Centre of a partially ordered set

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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=32614
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article