AXIOMATISABILITY OF WEAKLY FLAT, FLAT, AND PROJECTIVE<i>S</i>-ACTS

Bulman-Fleming, Sydney; Gould, Victoria

doi:10.1081/agb-120015672

Cited by 7 publications

(18 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We furnish a description of partially ordered monoids with an axiomatizable class of partially ordered flat, weakly flat, and strongly flat polygons. This result is similar to the appropriate result for polygons (see [2,3]). As in the unordered case, for a commutative partially ordered monoid, it is proved that completeness of a class of strongly flat partially ordered polygons over the monoid is equivalent to the class being model complete, categorical, and to the monoid being a partially ordered Abelian group.…”

supporting

confidence: 89%

“…(2) and ultrapower of S S satisfies (P < ); (3) for any s, t ∈ S, the set R < (s, t) either is empty or is finitely generated as a subpolygon of a right polygon (S × S) S . Combining Theorems 4.5 and 4.7, with Theorem 2.3 in mind, we obtain the following result for the class SF < (which is the counterpart of a result for the class of strongly flat polygons in [3]). (1) the class SF < is axiomatizable;…”

Section: Axiomatizability and Completeness Of The Class Of Flat Partimentioning

confidence: 65%

“…Here, those results are generalized to po-polygons. Lemma 4.1 and Theorems 4.2 and 4.3 are analogs of corresponding theorems for polygons in [3]. To prove these results, we need only introduce changes associated with features particular to po-polygons.…”

Section: Axiomatizability and Completeness Of The Class Of Flat Partimentioning

confidence: 89%

“…Monoids with an axiomatizable class of weakly flat and flat polygons are described in [3]. Here, those results are generalized to po-polygons.…”

Section: Axiomatizability and Completeness Of The Class Of Flat Partimentioning

confidence: 99%

See 3 more Smart Citations

Axiomatizability and completeness of some classes of partially ordered polygons

Pervukhin

Stepanova

2009

Algebra Logic

View full text Add to dashboard Cite

We study partially ordered monoids over which a class of free (over sets and over posets), projective, and (strongly, weakly) flat partially ordered polygons is axiomatizable, complete, or model complete. Similar issues for polygons were dealt with in papers by V. , classes of free, projective, and (strongly, weakly) flat polygons were explored in relation to their axiomatizability, completeness, and model completeness. In the present paper, we look into the same issues, but now for classes of partially ordered polygons. Here, the concept of a left perfect monoid is carried over the case of a partially ordered monoid, and certain conditions equivalent to this concept are formulated. We furnish a description of partially ordered monoids with an axiomatizable class of partially ordered flat, weakly flat, and strongly flat polygons. This result is similar to the appropriate result for polygons (see [2,3]). As in the unordered case, for a commutative partially ordered monoid, it is proved that completeness of a class of strongly flat partially ordered polygons over the monoid is equivalent to the class being model complete, categorical, and to the monoid being a partially ordered Abelian group. It is known that a partially ordered polygon is strongly flat iff it satisfies conditions (P < ) and (E < ) (see [4]).Here we study partially ordered monoids with axiomatizable, complete, model complete, and categorical classes of partially ordered polygons satisfying (P < ) or (E < ). The results obtained for axiomatizable, complete, model complete, and categorical classes of partially ordered projective polygons free over a set are similar to corresponding results for polygons (see [1,2]). Again, we describe partially ordered monoids with finitely many different right ideals over which a class of partially ordered polygons free over a poset is axiomatizable, and remark that there does not exist a partially ordered monoid for which this class is axiomatizable and complete.The results presented in Sec. 3 are due to Stepanova; Theorem 2.6 was proved jointly by the two authors; other results are due to Pervukhin.

show abstract

supporting

confidence: 89%

Section: Axiomatizability and Completeness Of The Class Of Flat Partimentioning

confidence: 65%

Section: Axiomatizability and Completeness Of The Class Of Flat Partimentioning

confidence: 89%

“…Monoids with an axiomatizable class of weakly flat and flat polygons are described in [3]. Here, those results are generalized to po-polygons.…”

Section: Axiomatizability and Completeness Of The Class Of Flat Partimentioning

confidence: 99%

See 2 more Smart Citations

Axiomatizability and completeness of some classes of partially ordered polygons

Pervukhin

Stepanova

2009

Algebra Logic

View full text Add to dashboard Cite

show abstract

“…We devote Sections 5 to 9 to characterising those monoids S such that the classes of weakly flat, flat, strongly flat, projective and free S-acts are axiomatisable. The material for Sections 5 to 8 is taken from the papers [10] and [2] of the first author and Bulman-Fleming, and the paper [22] of the fourth author. Section 9 contains a new result of the first author characterising those monoids such that the class of free left S-acts is axiomatisable, and some specialisations taken from [22].…”

Section: Introductionmentioning

confidence: 99%

Model-theoretic properties of free, projective, and flat S-acts

et al. 2009

View full text Add to dashboard Cite

This is the second in a series of articles surveying the body of work on the model theory of S-acts over a monoid S. The first concentrated on the theory of regular S-acts. Here we review the material on model-theoretic properties of free, projective and (strongly, weakly) flat S-acts. We consider questions of axiomatisability, completeness, model completeness and stability for these classes. Most but not all of the results have already appeared; we remark that the description of those monoids S such that the class of free left S-acts is axiomatisable, is new.

show abstract

Axiomatisability problems for S-posets

Gould

Shaheen

2010

Semigroup Forum

Self Cite

View full text Add to dashboard Cite

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.

show abstract

AXIOMATISABILITY OF WEAKLY FLAT, FLAT, AND PROJECTIVES-ACTS

Cited by 7 publications

References 15 publications

Axiomatizability and completeness of some classes of partially ordered polygons

Axiomatizability and completeness of some classes of partially ordered polygons

Model-theoretic properties of free, projective, and flat S-acts

Axiomatisability problems for S-posets

Contact Info

Product

Resources

About