Birget–rhodes Expansion of Groups and Inverse Monoids of Möbius Type

Choi, Keunbae; Lim, Yongdo

doi:10.1142/s0218196702001073

Cited by 7 publications

(5 citation statements)

References 5 publications

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“…However, Kellendonk and Lawson show, in one of the main results of [7], that S(G) is in fact the Birget-Rhodes expansion [1] of the group G. This description clarifies the structure of S(G), and adds further interest to Exel's approach which is now seen to give a presentation of the Birget-Rhodes expansion. Choi and Lim [3] use the results of [7] to describe the structure of the Birget-Rhodes expansions of certain groups acting on hausdorff spaces.…”

Section: Introductionmentioning

confidence: 99%

Actions and expansions of ordered groupoids

Gilbert

2005

Journal of Pure and Applied Algebra

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We construct the Birget-Rhodes expansion G BR of an ordered groupoid G. The construction is given in terms of certain finite subsets of G, but we also show how G BR can be considered as a prefix expansion. Moreover, for an inductive groupoid G we recover the prefix expansion of Lawson-Margolis-Steinberg. We show that the Birget-Rhodes expansion of an ordered groupoid G classifies partial actions of G on a set X: the correspondence between partial actions of G and actions of G BR can be viewed as a partial-to-global result achieved by enlarging the acting groupoid. We further discuss globalisation achieved by enlarging the set acted upon and show that a groupoid variant of the tensor product of G-sets provides a canonical globalisation of any partial action.We also sketch the construction of the Margolis-Meakin expansion (G, A) MM of an ordered groupoid G with generating set A. This is related to the Birget-Rhodes expansion and was first defined by Margolis and Meakin for a group G generated by A in terms of finite subgraphs of the Cayley graph (G, A).

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Section: Introductionmentioning

confidence: 99%

Actions and expansions of ordered groupoids

Gilbert

2005

Journal of Pure and Applied Algebra

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“…The first algebraic results on the subject appeared in [89,91,117,120,151,204,205,275,276], including the first algebraic application for tiling semigroups in [204] (with further use in [283]), for E-unitary inverse semigroups in [205], to inverse semigroups and F-inverse monoids in [275] and to inverse monoids of Möbius type in [91]. Independently from Exel's definition of a partial action, Coulbois [110] used a more restrictive notion, called a pre-action, to deal with the Ribes-Zalesski property (R Z n ) of groups from model theoretic point of view (see also [111]).…”

Section: Mathematics Subject Classificationmentioning

confidence: 99%

“…The above facts turned S(G) into a highly important tool, especially when dealing with partial projective group representations [129][130][131], and the cohomology theory based on partial actions [124][125][126], as well as when relating crossed products by partial actions of groups with crossed products by inverse semigroup actions [163]. They were used in [91] to study realizations of Pr(G) as an inverse monoid of Möbius type related to a partial action of G on a Hausdorff space. Moreover, the expansion method was further developed and used for partial actions of inverse semigroups in [68,220], of groupoids in [37,40,178], of monoids in [199], of restriction semigroups in [197] and of inductive constellations in [198].…”

Section: Mathematics Subject Classificationmentioning

confidence: 99%

Recent developments around partial actions

Dokuchaev

2018

São Paulo J. Math. Sci.

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We give an overview of publications on partial actions and related concepts, paying main attention to some recent developments on diverse aspects of the theory, such as partial actions of semigroups, of Hopf algebras and groupoids, the globalization problem for partial actions, Morita theory of partial actions, twisted partial actions, partial projective representations and the Schur multiplier, cohomology theories related to partial actions, Galois theoretic results, ring theoretic properties and ideals of partial crossed products. Among the applications we consider in more detail the case of the Carlsen-Matsumoto C *-algebra related to an arbitrary subshift, but also mention many others. The total number of publications directly related to partial actions and partial representations is more than 130, so that it is impossible even to describe briefly the content of all of them within the constraints of the present survey. Thus, the majority of them are only cited with respects to specific topics, trying to give an idea about the involved matter. In order to complete the picture, we refer the reader to a recent book by Ruy Exel, to our previous surveys, as well as to those by other authors.

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“…The first algebraic results on the subject appeared in [151], [120], [204], [275], [91], [276], [205], [89] and [117], including the first algebraic application for tiling semigroups in [204] (with further use in [283]), for E-unitary inverse semigroups in [205], to inverse semigroups and F -inverse monoids in [275] and to inverse monoids of Möbius type in [91]. Independently from Exel's definition of a partial action, T. Coulbois [110] used a more restrictive notion, called a pre-action, to deal with the Ribes-Zalesski property (RZ n ) of groups from model theoretic point of view (see also [111]).…”

mentioning

confidence: 99%

“…The above facts turned S(G) into a highly important tool, especially when dealing with partial projective group representations [129], [130], [131], and the cohomology theory based on partial actions [124], [125], [126], as well as when relating crossed products by partial actions of groups with crossed products by inverse semigroup actions [163]. They were used in [91] to study realizations of Pr(G) as an inverse monoid of Möbius type related to a partial action of G on a Hausdorff space. Moreover, the expansion method was further developed and used for partial actions of inverse semigroups in [220], [68], of groupoids in [178], [37], [40], of monoids in [199], of restriction semigroups in [197] and of inductive constellations in [198].…”

mentioning

confidence: 99%