Axiomatisability problems for S-acts, revisited

Axiomatisability problems for S-posets

“…Conditions (P) and (E) and the class SF . For completeness we give the following results from [15]; they may also be found in the thesis of the second author [16]. The proofs follow closely those of the unordered case in [10], [11] and [12].…”

“…Hence SF = Pr. From [15,Theorem 4.8] and [16], we also have that SF is axiomatisable. Hence Pr is axiomatisable.…”

“…This will enable us to specialise to the case where C is the class of all right S-poset embeddings. For the second we consider an arbitary class C; we then specialise to the cases where C consists of all inclusions of (principal) right ideals into S. We remark that similar methods can be applied to axiomatisability problems for S-acts over a monoid S, as shown in [16].…”

“…The statements of these results are obtained from those of Corollaries 3.10 and 3.11, with the word 'double' omitted and 'flat' replaced by 'poflat'. Further details may be found in [16].…”

“…Corresponding questions for classes of M-acts over a monoid M have been answered in [10,19,1] and [11], see also the survey article [12]. The classes of projective (strongly flat, po-flat, weakly po-flat) left S-posets Pr(SF , PF , WPF ) have recently been considered in [15] (which uses slightly different terminology; the results also appearing in [16]) as has the class F r of free left S-posets in the case where S has only finitely many right ideals. We note that many of the techniques of [15] follow those in the M-act case and, for this reason, we aim here to produce two general strategies that will deal with a number of axiomatisability questions for classes of S-posets (and, with minor adjustment, M-acts).…”