BIRGET–RHODES EXPANSION OF GROUPS AND INVERSE MONOIDS OF MÖBIUS TYPE

2002 ◽  
Vol 12 (04) ◽  
pp. 525-533 ◽  
Author(s):  
KEUNBAE CHOI ◽  
YONGDO LIM
Keyword(s):  
Hausdorff Space ◽  
Inverse Monoid ◽  
Expansion Formula ◽  
Inverse Monoids ◽  
Isolated Point

In this paper we prove that if a group G acts faithfully on a Hausdorff space X and acts freely at a non-isolated point, then the Birget–Rhodes expansion [Formula: see text] of the group G is isomorphic to an inverse monoid of Möbius type obtained from the action.

Two-sided expansions of monoids

2019 ◽  
Vol 29 (08) ◽  
pp. 1467-1498 ◽  
Author(s):  
Ganna Kudryavtseva
Keyword(s):  
Inverse Monoid ◽  
Partial Action ◽  
Expansion Formula ◽  
Inverse Monoids ◽  
Universal Formula ◽  
Special Case ◽  
Free Inverse Monoid

We initiate the study of expansions of monoids in the class of two-sided restriction monoids and show that generalizations of the Birget–Rhodes prefix group expansion, despite the absence of involution, have rich structure close to that of relatively free inverse monoids. For a monoid [Formula: see text] and a class of partial actions of [Formula: see text], determined by a set, [Formula: see text], of identities, we define [Formula: see text] to be the universal [Formula: see text]-generated two-sided restriction monoid with respect to partial actions of [Formula: see text] determined by [Formula: see text]. This is an [Formula: see text]-restriction monoid which (for a certain [Formula: see text]) generalizes the Birget–Rhodes prefix expansion [Formula: see text] of a group [Formula: see text]. Our main result provides a coordinatization of [Formula: see text] via a partial action product of the idempotent semilattice [Formula: see text] of a similarly defined inverse monoid, partially acted upon by [Formula: see text]. The result by Fountain, Gomes and Gould on the structure of the free two-sided restriction monoid is recovered as a special case of our result. We show that some properties of [Formula: see text] agree well with suitable properties of [Formula: see text], such as being cancellative or embeddable into a group. We observe that if [Formula: see text] is an inverse monoid, then [Formula: see text], the free inverse monoid with respect to strong premorphisms, is isomorphic to the Lawson–Margolis–Steinberg generalized prefix expansion [Formula: see text]. This gives a presentation of [Formula: see text] and leads to a model for [Formula: see text] in terms of the known model for [Formula: see text].

FREE INVERSE MONOIDS AND GRAPH IMMERSIONS

1993 ◽  
Vol 03 (01) ◽  
pp. 79-99 ◽  
Author(s):  
STUART W. MARGOLIS ◽  
JOHN C. MEAKIN
Keyword(s):  
Formal Language ◽  
Inverse Monoid ◽  
Finitely Generated ◽  
Inverse Monoids ◽  
Rational Subset ◽  
The Relationship ◽  
Subgroup Theorem ◽  
Free Inverse Monoid ◽  
Natural Way

The relationship between covering spaces of graphs and subgroups of the free group leads to a rapid proof of the Nielsen-Schreier subgroup theorem. We show here that a similar relationship holds between immersions of graphs and closed inverse submonoids of free inverse monoids. We prove using these methods, that a closed inverse submonoid of a free inverse monoid is finitely generated if and only if it has finite index if and only if it is a rational subset of the free inverse monoid in the sense of formal language theory. We solve the word problem for the free inverse category over a graph Γ. We show that immersions over Γ may be classified via conjugacy classes of loop monoids of the free inverse category over Γ. In the case that Γ is a bouquet of X circles, we prove that the category of (connected) immersions over Γ is equivalent to the category of (transitive) representations of the free inverse monoid FIM(X). Such representations are coded by closed inverse submonoids of FIM(X). These monoids will be constructed in a natural way from groups acting freely on trees and they admit an idempotent pure retract onto a free inverse monoid. Applications to the classification of finitely generated subgroups of free groups via finite inverse monoids are developed.

Inverse semigroup homomorphisms via partial group actions

2001 ◽  
Vol 64 (1) ◽  
pp. 157-168 ◽  
Author(s):  
Benjamin Steinberg
Keyword(s):  
Inverse Semigroup ◽  
Group Actions ◽  
Inverse Semigroups ◽  
Inverse Monoid ◽  
Partial Group ◽  
Inverse Monoids ◽  
Maximal Group ◽  
Special Cases ◽  
Quick Proof ◽  
Group Component

This papar constructs all homomorphisms of inverse semigroups which factor through an E-unitary inverse semigroup; the construction is in terms of a semilattice component and a group component. It is shown that such homomorphisms have a unique factorisation βα with α preserving the maximal group image, β idempotent separating, and the domain I of β E-unitary; moreover, the P-representation of I is explicitly constructed. This theory, in particular, applies whenever the domain or codomain of a homomorphism is E-unitary. Stronger results are obtained for the case of F-inverse monoids.Special cases of our results include the P-theorem and the factorisation theorem for homomorphisms from E-unitary inverse semigroups (via idempotent pure followed by idempotent separating). We also deduce a criterion of McAlister–Reilly for the existence of E-unitary covers over a group, as well as a generalisation to F-inverse covers, allowing a quick proof that every inverse monoid has an F-inverse cover.

scholarly journals Construction of Inverse Unit Regular Monoids from a Semilattice and a Group

2018 ◽  
Vol 7 (4.36) ◽  
pp. 950
Author(s):  
Sreeja V.K
Keyword(s):  
Inverse Monoid ◽  
Inverse Monoids ◽  
Regular Semigroups ◽  
Group Of Units ◽  
Regular Monoid

This paper is a continuation of a previous paper [6] in which the structure of certain unit regular semigroups called R-strongly unit regular monoids has been studied. A monoid S is said to be unit regular if for each element s Î S there exists an element u in the group of units G of S such that s = sus. Hence where su is an idempotent and is a unit. A unit regular monoid S is said to be a unit regular inverse monoid if the set of idempotents of S form a semilattice. Since inverse monoids are R unipotent, every element of a unit regular inverse monoid can be written as s = eu where the idempotent part e is unique and u is a unit. Here we give a detailed study of inverse unit regular monoids and the results  are mainly based on [10]. The relations between the semilattice of idempotents and the group of units in unit regular inverse monoids are better identified in this case. .

scholarly journals Inverse monoids and immersions of 2-Complexes

2015 ◽  
Vol 25 (01n02) ◽  
pp. 301-323 ◽  
Author(s):  
John Meakin ◽  
Nóra Szakács
Keyword(s):  
Topological Space ◽  
Fundamental Group ◽  
Conjugacy Classes ◽  
Inverse Monoid ◽  
Two Dimensional ◽  
Inverse Monoids ◽  
Mild Conditions ◽  
Cw Complex ◽  
Cw Complexes

It is well known that under mild conditions on a connected topological space 𝒳, connected covers of 𝒳 may be classified via conjugacy classes of subgroups of the fundamental group of 𝒳. In this paper, we extend these results to the study of immersions into two-dimensional CW-complexes. An immersion f : 𝒟 → 𝒞 between CW-complexes is a cellular map such that each point y ∈ 𝒟 has a neighborhood U that is mapped homeomorphically onto f(U) by f. In order to classify immersions into a two-dimensional CW-complex 𝒞, we need to replace the fundamental group of 𝒞 by an appropriate inverse monoid. We show how conjugacy classes of the closed inverse submonoids of this inverse monoid may be used to classify connected immersions into the complex.

Types over c(k) spaces

2004 ◽  
Vol 77 (1) ◽  
pp. 17-28
Author(s):  
Markus Pomper
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Lower Semicontinuous ◽  
Continuous Functions ◽  
Upper Semicontinuous ◽  
Bounded Functions ◽  
Isolated Point

AbstractLet K be a compact Hausdorff space and C(K) the Banach space of all real-valued continuous functions on K, with the sup norm. Types over C(K) (in the sense of Krivine and Maurey) are represented here by pairs (l, u) of bounded real-valued functions on K, where l is lower semicontinuous and u is upper semicontinuous, l ≤ u and l(x) = u(x) for every isolated point x of K. For each pair the corresponding type is defined by the equation τ(g) = max{║l + g║∞, ║u + g║∞} for all g ∈ C(K), where ║·║∞ is the sup norm on bounded functions. The correspondence between types and pairs (l, u) is bijective.

scholarly journals Unit regular inverse monoids and Cliford monoids

2018 ◽  
Vol 7 (2.13) ◽  
pp. 306
Author(s):  
Sreeja V K
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Group Element ◽  
Inverse Monoid ◽  
Completely Regular Semigroup ◽  
Clifford Semigroup ◽  
Inverse Monoids ◽  
Completely Regular ◽  
Group Of Units

Let S be a unit regular semigroup with group of units G = G(S) and semilattice of idempotents E = E(S). Then for every there is a such that Then both xu and ux are idempotents and we can write or .Thus every element of a unit regular inverse monoid is a product of a group element and an idempotent. It is evident that every L-class and every R-class contains exactly one idempotent where L and R are two of Greens relations. Since inverse monoids are R unipotent, every element of a unit regular inverse monoid can be written as s = eu where the idempotent part e is unique and u is a unit. A completely regular semigroup is a semigroup in which every element is in some subgroup of the semigroup. A Clifford semigroup is a completely regular inverse semigroup. Characterization of unit regular inverse monoids in terms of the group of units and the semilattice of idempotents is a problem often attempted and in this direction we have studied the structure of unit regular inverse monoids and Clifford monoids. 

Inverse monoids and immersions of Δ-complexes

2021 ◽  
pp. 1-26
Author(s):  
John Meakin ◽  
Nóra Szakács
Keyword(s):  
Fundamental Group ◽  
Conjugacy Classes ◽  
Base Space ◽  
Inverse Monoid ◽  
Inverse Monoids ◽  
Finite Dimensional

An immersion [Formula: see text] between [Formula: see text]-complexes is a [Formula: see text]-map that induces injections from star sets of [Formula: see text] to star sets of [Formula: see text]. We study immersions between finite-dimensional connected [Formula: see text]-complexes by replacing the fundamental group of the base space by an appropriate inverse monoid. We show how conjugacy classes of the closed inverse submonoids of this inverse monoid may be used to classify connected immersions into the complex. This extends earlier results of Margolis and Meakin for immersions between graphs and of Meakin and Szakács on immersions into 2-dimensional [Formula: see text]-complexes.

scholarly journals ON FREE SPECTRA OF A CLASS OF FINITE INVERSE MONOIDS

2012 ◽  
Vol 93 (3) ◽  
pp. 225-237
Author(s):  
IGOR DOLINKA
Keyword(s):  
Semidirect Product ◽  
Natural Order ◽  
Inverse Monoid ◽  
Left Action ◽  
Inverse Monoids ◽  
Group Of Units ◽  
Finite Monoids ◽  
Free Spectrum ◽  
Free Spectra

AbstractFor a finite Clifford inverse algebra $A$, with natural order meet-semilattice ${Y}_{A} $ and group of units ${G}_{A} $, we show that the inverse monoid obtained as the semidirect product ${ Y}_{A}^{1} {\mathop{\ast }\nolimits}_{\rho } {G}_{A} $ has a log-polynomial free spectrum whenever $\rho $ is a term-expressible left action of ${G}_{A} $ on ${Y}_{A} $ and all subgroups of $A$ are nilpotent. This yields a number of examples of finite inverse monoids satisfying the Seif conjecture on finite monoids whose free spectra are not doubly exponential.

A CORRESPONDENCE BETWEEN BALANCED VARIETIES AND INVERSE MONOIDS

2006 ◽  
Vol 16 (05) ◽  
pp. 887-924 ◽  
Author(s):  
MARK V. LAWSON
Keyword(s):  
Special Class ◽  
Linear Equations ◽  
Special Kind ◽  
Free Monoid ◽  
Inverse Monoid ◽  
Inverse Monoids ◽  
Group V ◽  
Thompson's Groups ◽  
Term Algebra ◽  
Thompson Group

There is a well-known correspondence between varieties of algebras and fully invariant congruences on the appropriate term algebra. A special class of varieties are those which are balanced, meaning they can be described by equations in which the same variables appear on each side. In this paper, we prove that the above correspondence, restricted to balanced varieties, leads to a correspondence between balanced varieties and inverse monoids. In the case of unary algebras, we recover the theorem of Meakin and Sapir that establishes a bijection between congruences on the free monoid with n generators and wide, positively self-conjugate inverse submonoids of the polycyclic monoid on n generators. In the case of varieties generated by linear equations, meaning those equations where each variable occurs exactly once on each side, we can replace the clause monoid above by the linear clause monoid. In the case of algebras with a single operation of arity n, we prove that the linear clause monoid is isomorphic to the inverse monoid of right ideal isomorphisms between the finitely generated essential right ideals of the free monoid on n letters, a monoid previously studied by Birget in the course of work on the Thompson group V and its analogues. We show that Dehornoy's geometry monoid of a balanced variety is a special kind of inverse submonoid of ours. Finally, we construct groups from the inverse monoids associated with a balanced variety and examine some conditions under which they still reflect the structure of the underlying variety. Both free groups and Thompson's groups Vn,1 arise in this way.

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