Axiomatisability problems for S-posets

Abstract

Let C be a class of algebras of a given fixed type τ . Associated with the type is a first order language L τ . One can then ask the question, when is the class C axiomatisable by sentences of L τ ? 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.

Some more axiomatisability for S-acts

“…We present here a more general procedure to axiomatise these classes. A similar type of general results have been found for S-posets by Gould and Shaheen [10]. We have found that there are some classes of S-acts which are axiomatisable by more than one method.…”

“…We denote the class of C-flat left S-act by CF . Note: for S-posets there will be two variations of the notion of C-flat, as we explain in [10].…”

“…Proof. The proof is along the similar lines as given for ordered case in [10], for class EP ≤ , or see detail proof in [13].…”

“…The aim of this paper is to add the theory of axiomatisability of classes of S-acts over a monoid S. We put some of the techniques of earlier articles into a general setting. In [10] we use these methods to develop the theory of axiomatisability of S-posets over a pomonoid S.…”

“…These can then be specialised to prove both new and known results. For clarity we are giving general results without proves which can be solved on similar lines as for S-posets case, see [10]. Interested reader may find proves in [13].…”