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 $B'=A$.

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  • $\begingroup$ 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 $(A_1 \cap A_2, (B_1 \cup B_2)'')$ ? $\endgroup$ – Amrith Krishna Jan 3 at 14:03
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    $\begingroup$ @AmrithKrishna Yes, since $(B_1 \cup B_2)'=B_1' \cap B_2'$, as always in a Galois connection. $\endgroup$ – amrsa Jan 3 at 14:06

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