Need confirmation in a Hasse Diagram about Complement of an element in Lattice

Question

Following hasse diagram is Lattice. 

Is it right that complement of b – c only and complement of c – b only ?

Answer 1

If two elements a & $a^{c}$ are complement of each other, then they should satisfy :

$a \vee a^{c}$ = Upper Bound of Lattice

$a \wedge a^{c}$ = Lower Bound of Lattice

 

Here, the upper bound of lattice is element ‘d’ & lower bound is element ‘a’.

 

For element ‘b’ :

$b \vee c = d$ (Upper Bound of lattice)

$b \wedge c$ = a (Lower Bound of Lattice).

So, element ‘c’ is complement of ‘b’.

 

Similarly, for element ‘c’ :

$c \vee b = d$ (Upper Bound of lattice)

$c \wedge b$ = a (Lower Bound of Lattice).

So, element ‘b’ is complement of ‘c’.

 

So, element ‘b’ & element ‘c’ are complement of each other.