Let C be a class of algebras of a given fixed type t. Associated with the
type is a first order language L_t. One can then ask the question, when is the
class C axiomatisable by sentences of L_t. In this paper we will be considering
axiomatisability problems for classes of left S-posets over a pomonoid S (that
is, a monoid S equipped with a partial order compatible with the binary
operation). We aim to determine the pomonoids S such that certain categorically
defined classes are axiomatisable.
The classes we consider are the free S-posets, the projective S-posets and
classes arising from flatness properties. Some of these cases have been studied
in a recent article by Pervukhin and Stepanova. We present some general
strategies to determine axiomatisability, from which their results for the
classes of weakly po-flat and po-flat S-posets will follow. We also consider a
number of classes not previously examined
