# Understanding infimum in a complete lattice

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 $$B'=A$$.
• That would make it, $(A_1 \cap A_2, (B_1' \cap B_2' )')$. Are you saying by closure properties (or is it demorgan's law) it would result in $(A_1 \cap A_2, (B_1 \cup B_2)'')$ ? – Amrith Krishna Jan 3 at 14:03
• @AmrithKrishna Yes, since $(B_1 \cup B_2)'=B_1' \cap B_2'$, as always in a Galois connection. – amrsa Jan 3 at 14:06