Perfection for pomonoids

Gould, Victoria; Shaheen, Lubna

doi:10.1007/s00233-010-9237-y

Cited by 14 publications

(12 citation statements)

References 19 publications

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“…The current authors have shown that a pomonoid S is left perfect as a monoid if and only if it is left perfect as a pomonoid [13]. With this in mind we can give a short and direct proof of the following.…”

Section: Axiomatisability Of Projective and Free S-posetsmentioning

confidence: 67%

Axiomatisability problems for S-posets

Gould

Shaheen

2010

Semigroup Forum

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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.

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Section: Axiomatisability Of Projective and Free S-posetsmentioning

confidence: 67%

Axiomatisability problems for S-posets

Gould

Shaheen

2010

Semigroup Forum

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“…The concept of a unitary subsemigroup has been known to be related to the question of embeddability of amalgams since Howie's pioneering work in [8]. Gould and Shaheen [7] generalised this concept for posemigroups during their study of projective covers of pomonoids.…”

Section: Pounitary Subpomonoids and Amalgmationmentioning

confidence: 99%

“…LSRP U ∧ U SRP UThe implications are strict as the following examples demonstrate. In[7] it is shown that if a pomonoid U is right unitary in S then it need not be right pounitary in S. Let U = {0, 1, 2, 3, . .…”

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confidence: 99%

On Free Products and Amalgams of Pomonoids

Subaiei

Renshaw

2016

Communications in Algebra

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The study of amalgamation in the category of partially ordered monoids was initiated by Fakhuruddin in the 1980s. In 1986 he proved that, in the category of commutative pomonoids, every absolutely flat commutative pomonoid is a weak amalgmation base and every commutative pogroup is a strong amalgamation base. Some twenty years later, Bulman-Fleming and Sohail in 2011 extended this work to what they referred to as pomonoid amalgams. In particular they proved that pogroups are poamalgmation bases in the category of pomonoids. Sohail, also in 2011, proved that absolutely poflat commutative pomonoids are poamalgmation bases in the category of commutative pomonoids. In the present paper we extend the work on pomonoid amalgams by generalising the work of Renshaw on amalgams of monoids and extension properties of acts over monoids.

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“…In [7] the ordered version of locally cyclic acts is called a weakly locally cyclic S -poset as an Sposet A such that every finitely generated S -subposet of A is contained in a cyclic S -poset. Moreover, a principal left ideal of S that is also weakly locally cyclic is called weakly locally principal left ideal.…”

Section: Conditions (P) and (P W ) And Strongly Flatmentioning

confidence: 99%

“…The reader is referred to the monograph [8] for a complete discussion of flatness properties and definition of acts over monoids. On the other hand, the investigation of S -posets was initiated by Fakhruddin in the 1980s, and recently many papers on this topic have appeared, mostly concentrating on projectivity and various notions of flatness for S -posets, such as [5,3,7,11,10]. Following Section 1, we…”

Section: Introductionmentioning

confidence: 99%