Re: Poset adjoints

["Fred E.J. Linton" <fejlinton@usa.net>]

Properly posed, Vasili's question should be not

>        Given that poset adjoints are considered miniature versions of
> adjoints, what are are the unit and co-unit natural transformations
> say between poset A and B?

but:
  
: Given order-preserving functions r: A --> B and l: B --> A between
: posets A and B, with r right adjoint to l, what are the unit and 
: co-unit natural transformations 1_B ==> rl and lr ==> 1_A ?

And the answer, as always, is that 1_B: b --> rlb (for b in B) is 
the order relation b < rlb that adjointness correlates to lb < lb, 
while 1_A: lra --> a (for a in A) is the order relation lra < a 
that adjointness correlates to ra < ra.

(Above I'm writing simply < for the (reflexive, transitive) order relation
on either of the two posets.)

HTH. Cheers, -- Fred




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