I have a definition in a book:
a lattice $T$ can be complemented, if :
- it has one universal element $U$ and one null $n$,
- for each $x$ of $T$ can be associated at least one $\bar{x}$ of $T$, where $\left\{ \begin{array}{ll} x \lor \overline{x} = U \\ x \land \overline{x} = n \\ \end{array} \right.$
Being a computer engineer, I understand this well.
What I can't figure, and for which I would like to have an example from you, is :
What could be a lattice $T$ that :
- would have one universal element $U$ and one null $n$,
- but (for each $x$ of $T$ can be associated at least one $\bar{x}$ of $T$, where $\left\{ \begin{array}{ll} x \lor \overline{x} = U \\ x \land \overline{x} = n \\ \end{array} \right.$) would not be verified ?
Do you know an example, so that I can figure this myself clearer?