Complemented lattice

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In the mathematical discipline of order theory, and in particular, in lattice theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element a has a complement, i.e. an element b such that

a ∨ b = 1     and    a ∧ b = 0.

In general an element may have more than one complement. However, in a bounded distributive lattice every element will have at most one complement.^[1] A lattice in which every element has exactly one complement is called a uniquely complemented lattice.

A lattice with the property that every interval is complemented is called a relatively complemented lattice.

An orthocomplemented lattice is a lattice equipped with an involutive order-reversing function from elements to complements. A distributive orthocomplemented lattice is called a Boolean algebra.

Some complemented lattices

In the pentagon lattice N5, the node on the right-hand side has two complements.

The diamond lattice M3 admits no orthocomplementation.

The lattice M4 admits 3 orthocomplementations.

The hexagon lattice admits a unique orthocomplementation, but it is not uniquely complemented.

Notes

1. ^ Rutherford (1965), Th.9.3 p.25.

References

• Rutherford, Daniel Edwin (1965). Introduction to Lattice Theory. Oliver and Boyd.