For any two formal concepts, $(A_1,B_1)$ and $(A_2,B_2)$ of a formal context, the standard definition for the supremum and infimum in a complete lattice are as follows:
Greatest common subconcept or infimum: $$ (A_1, B_1) \wedge (A_2, B_2) := (A_1 \cap A_2, (B_1 \cup B_2)'') $$
Least common superconcept or Supremum: $$ (A_1, B_1) \vee (A_2, B_2) := ((A_1 \cup A_2)'', B_1 \cap B_2) $$
But alternatively, I have found the following definition for infimum as well:
$$ (A_1, B_1) \wedge (A_2, B_2) := (A_1 \cap A_2, (A_1 \cap A_2)') $$
This was in the context of a residuated lattice (which is a complete lattice as well), as defined in this paper by Alexander Clark (page 3, last paragraph)
Are these 2 definitions of infimum equivalent in the case of complete lattices?
It just follows from the fact that if $(A,B)$ is a formal concept, then $A'=B$ and $B'=A$.