Partial Actions of Monoids

Hollings, Christopher

doi:10.1007/s00233-006-0665-7

Cited by 34 publications

(67 citation statements)

References 12 publications

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“…The latter result was generalised to restriction semigroups by Lawson [14] (though he used different terminology). Lawson's approach was made explicit for restriction semigroups in Hollings [9]. Moreover, the ordering ≤ from the category coincides with the natural partial order of the restriction semigroup and ⊗ coincides with · whenever it is defined.…”

Section: Lemma 21 Let T Be a Semigroup Acting On The Left Of A Left mentioning

confidence: 99%

Restriction semigroups and $$\lambda $$ λ -Zappa-Szép products

Gould

Zenab

2016

Period Math Hung

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The aim of this paper is to study λ-semidirect and λ-Zappa-Szép products of restriction semigroups. The former concept was introduced for inverse semigroups by Billhardt, and has been extended to some classes of left restriction semigroups. The latter was introduced, again in the inverse case, by Gilbert and Wazzan. We unify these concepts by considering what we name the scaffold of a Zappa-Szép product S T where S and T are restriction. Under certain conditions this scaffold becomes a category. If one action is trivial, or if S is a semilattice and T a monoid, the scaffold may be ordered so that it becomes an inductive category. A standard technique, developed by Lawson and based on the Ehresmann-ScheinNambooripad result for inverse semigroups, allows us to define a product on our category. We thus obtain restriction semigroups that are λ-semidirect products and λ-Zappa-Szép products, extending the work of Billhardt and of Gilbert and Wazzan. Finally, we explicate the internal structure of λ-semidirect products.

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Section: Lemma 21 Let T Be a Semigroup Acting On The Left Of A Left mentioning

confidence: 99%

Restriction semigroups and $$\lambda $$ λ -Zappa-Szép products

Gould

Zenab

2016

Period Math Hung

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“…The notion of partial action which we will adopt is adapted from that of [13], in which can be found two further notions of partial monoid action. The first of these, a weak partial action [13 Each of the definitions of [13] can, of course, be applied to any monoid; however, in the particular case of a weakly left E-ample monoid, any results obtained by using these definitions (in particular, the results of [13]) will not, in general, respect the + operation.…”

Section: Partial Actions and Premorphismsmentioning

confidence: 99%

“…The first of these, a weak partial action [13 Each of the definitions of [13] can, of course, be applied to any monoid; however, in the particular case of a weakly left E-ample monoid, any results obtained by using these definitions (in particular, the results of [13]) will not, in general, respect the + operation. Therefore, for weakly left E-ample monoids, we choose to augment both definitions of partial action (as well as the definition of an incomplete action) with an additional axiom which reflects the presence of + .…”

Section: Partial Actions and Premorphismsmentioning

confidence: 99%

“…In Section 6, we will find it necessary temporarily to reinstate axiom (PA1) for strong partial actions; we will then refer to strong unitary partial actions. In the interests of consistency, we retain the numbering of [13]. DEFINITION 3.3 (Weak partial action).…”

Section: Partial Actions and Premorphismsmentioning

confidence: 99%

“…The study of the partial actions of monoids was taken up in [13], in which it was shown that the partial monoid actions of Megrelishvili and Schröder (termed strong…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

PARTIAL ACTIONS OF INVERSE AND WEAKLY LEFT E-AMPLE SEMIGROUPS

Gould

Hollings

2009

J. Aust. Math. Soc.

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We introduce partial actions of weakly left E-ample semigroups, thus extending both the notion of partial actions of inverse semigroups and that of partial actions of monoids. Weakly left E-ample semigroups arise very naturally as subsemigroups of partial transformation semigroups which are closed under the unary operation α → α + , where α + is the identity map on the domain of α. We investigate the construction of 'actions' from such partial actions, making a connection with the FA-morphisms of Gomes. We observe that if the methods introduced in the monoid case by Megrelishvili and Schröder, and by the second author, are to be extended appropriately to the case of weakly left E-ample semigroups, then we must construct not global actions, but so-called incomplete actions. In particular, we show that a partial action of a weakly left E-ample semigroup is the restriction of an incomplete action. We specialize our approach to obtain corresponding results for inverse semigroups.2000 Mathematics subject classification: primary 20M30; secondary 20M18.

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Relations in Mathematical Morphology with Applications to Graphs and Rough Sets

Stell

Lecture Notes in Computer Science

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Partial Actions of Monoids

Cited by 34 publications

References 12 publications

Restriction semigroups and $$\lambda $$ λ -Zappa-Szép products

Restriction semigroups and $$\lambda $$ λ -Zappa-Szép products

PARTIAL ACTIONS OF INVERSE AND WEAKLY LEFT E-AMPLE SEMIGROUPS

Relations in Mathematical Morphology with Applications to Graphs and Rough Sets

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