Subpullback flat \(S\)-posets need not be subequalizer flat. (Q1014775)

A right \(S\)-poset \(A_S\) is called `subpullback flat' (resp. `subequalizer flat') if the functor \(A_S\otimes-\) preserves subpullbacks (resp. subequalizers). It is proved that if \(S\) is a non-trivial, commutative, cancellative pomonoid in which 1 is the greatest element, then the one-element \(S\)-poset \(\Theta_S\) is subpullback flat but is not subequalizer flat. A new proof of the fact that a pullback flat \(S\)-act is equalizer flat is presented as well.