View source for 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.