I am taking a course of Introduction to Formal Concept Analysis and I have an uncertainty about the definition of supremum (least comum superconcept) and infimum (greatest comum subconcept) of formal concepts:
$(A_1,B_1)\wedge(A_2,B_2)=(A_1\cap A_2,(B_1\cup B_2)'')$
$(A_1,B_1)\vee(A_2,B_2)=((A_1\cup A_2)'',B_1\cap B_2)$
Why get we the double prime (double quote)?
The professor of the lecture has said
"because intents must be closed, we take the closure".
However, the first case is about extent, or does not...? I'm a little confused...
And we have to have both extent and intent closure, right?
Why just the unions have the closures? Maybe intersections of closed sets are closed, but unions not always...? Could any explication here...?
Doing exercises I feel the necessity of the closures to unions, but cannot draw yet a formal proof.
I'm with difficult in researching about this because almost all the results are about closed and open sets (topology), what are a little less general that this theory of the course that I'm taking.
Remembering that as I say 'the set $A$ is closed' I mean $A=A''$.
Thanks very much.
I tried again and I guess now I could understand... The intersections of intensions (similar to extents) are always an intent. Really, the set of the intensions (similar to extents) in a formal context in a closure system. However, this is not true for the union. To prove it we can get contraexamples (and there are plenty of these). And it's enough... What do you think?
PS.: Someone know how do we read $\wedge$ and $\vee$ in portuguese?