A bounded lattice for which every element has a complement is called a complemented lattice.
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A lattice in which every element has exactly one complement is called a "'uniquely complemented lattice "'
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The lattice of subspaces of a vector space provide an example of a complemented lattice that is not, in general, distributive.
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The groups whose lattice of subgroups is a complemented lattice are called complemented groups, and the groups whose lattice of subgroups are modular lattices are called Iwasawa groups or modular groups.
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The notation of 0 and 1 is used preferably when the poset is even a complemented lattice, and when no confusion is likely, i . e . when one is not talking about partial orders of numbers that already contain elements 0 and 1 different from bottom and top.
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Maltsev's characterization explains a large number of similar results in groups ( take ), rings, quasigroups ( take, complemented lattices, Heyting algebras etc . Furthermore, every congruence-permutable algebra is congruence-modular, i . e . its lattice of congruences is modular lattice as well; the converse is not true however.
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