View source for Centre of a partially ordered set

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.