Subpullback flat S-posets need not be subequalizer flat

Abstract

If S is a monoid, a right S-act A _S is a set A, equipped with a “right S-action” A×S→A sending the pair (a,s)∈ A×S to as, that satisfies the conditions (i) a(st)=(as)t and (ii) a1=a for all a∈A and s,t∈S. If, in addition, S is equipped with a compatible partial order and A is a poset, such that the action is monotone (when A×S is equipped with the product order), then A _S is called a right S-poset. Left S-acts and S-posets are defined analogously. For a given S-act (resp. S-poset) a tensor product functor A _S ⊗− from left S-acts to sets (resp. left S-posets to posets) exists, and A _S is called pullback flat or equalizer flat (resp. subpullback flat or subequalizer flat) if this functor preserves pullbacks or equalizers (resp. subpullbacks or subequalizers). By analogy with the Lazard-Govorov Theorem for R-modules, B. Stenström proved in 1971 that an S-act is isomorphic to a directed colimit of finitely generated free S -acts if and only if it is both pullback flat and equalizer flat. Some 20 years later, the present author showed that, in fact, pullback flatness by itself is sufficient. (A new, more direct proof of that result is contained in the present article.) In 2005, Valdis Laan and the present author obtained a version of the Lazard-Govorov Theorem for S-posets, in which subpullbacks and subequalizers now assume the role previously played by pullbacks and equalizers. The question of whether subpullback flatness implies subequalizer flatness remained unsolved. The present paper provides a negative answer to this question.