Definition:Uniquely Complemented Lattice

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Definition

Let $\left({S,\wedge,\vee,\preceq}\right)$ be a complemented lattice.


If each element of $S$ has only one complement, then $\left({S,\wedge,\vee,\preceq}\right)$ is a uniquely complemented lattice.

That is, a uniquely complemented lattice is a lattice in which each element has exactly one complement.


Note

Each uniquely complemented lattice supports a complement operation.

If $a$ is an element of the underlying set of a uniquely complemented lattice, we denote its complement by $\neg a$.

Note that some writers will denote it by $a'$.