In mathematics, an inverse semigroup (occasionally called an inversion semigroup) S is a semigroup in which every element x in S has a unique inversey in S in the sense that x = xyx and y = yxy, i.e. a regular semigroup in which every element has a unique inverse. Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of partial symmetries.
(The convention followed in this article will be that of writing a function on the right of its argument, and
composing functions from left to right — a convention often observed in semigroup theory.)
Project Name: Mathematical sciences without walls
Project Investigator: Dr. R. Ramanujam
Module Name: Inverse semigroups and the Cuntz-Li algebras
published: 03 Jun 2015
Inverse semigroup theory of cutting planes for integer linear optimization Part 2
Optimization Seminar: Prof. Matthias Koeppe from University of California, Davis
Title: Inverse semigroup theory of cutting planes for integer linear optimization
Abstract: MIP practitioners can solve large-scale mixed integer optimizationproblems to optimality or near-optimality by competent modelizationand use of branch-and-cut solvers. This technology was enabled to alarge part by the revival of Gomory’s classic general-purpose cuttingplanes such as the Gomory mixed integer cut. In the theory of such general-purpose cutting planes (validinequalities) the traditional, finite-dimensional techniques ofpolyhedral combinatorics are complemented by infinite-dimensionalmethods, the study of cut-generating functions. In my talk I will introduce the classic Gomory-Johnson model, auniversal r...
published: 11 Feb 2020
Inverse semigroup theory of cutting planes for integer linear optimization Part 1
Optimization Seminar: Prof. Matthias Koeppe from University of California, Davis
Title: Inverse semigroup theory of cutting planes for integer linear optimization
Abstract: MIP practitioners can solve large-scale mixed integer optimizationproblems to optimality or near-optimality by competent modelizationand use of branch-and-cut solvers. This technology was enabled to alarge part by the revival of Gomory’s classic general-purpose cuttingplanes such as the Gomory mixed integer cut. In the theory of such general-purpose cutting planes (validinequalities) the traditional, finite-dimensional techniques ofpolyhedral combinatorics are complemented by infinite-dimensionalmethods, the study of cut-generating functions. In my talk I will introduce the classic Gomory-Johnson model, auniversal re...
published: 11 Feb 2020
Inverse semigroups and the cuntz li algebras
Course : Mathematical sciences without walls
published: 12 May 2021
Nora Szakacs: Inverse semigroups as metric spaces, and their uniform Roe algebras
Recorded during Group Theory Seminar the February 22, 2023 at ENS, Paris
published: 20 Mar 2023
Amenability versus paradoxical decompositions for inverse semigroups and their C*-algebras
Speaker:
Fernando Lledó, University Carlos III de Madrid and ICMAT Madrid.
Event:
The 48th Canadian Operator Symposium
http://www.fields.utoronto.ca/activities/19-20/COSy
Title: Amenability versus paradoxical decompositions for inverse semigroups and their C*-algebras
In this talk I will address the dichotomy between amenability and the existence of paradoxical decompositions for inverse semigroups. Semigroups have singular dynamics in the sense that multiplication by an element can drastically shrink the size of a finite subset in the semigroup. This fact forces to analyze again basic aspects coming from group theory like paradoxical decompositions, Foelner nets or existence of invariant means. We will analyze these aspects in relation to two C*-algebras that generalize the usual...
published: 23 Jun 2020
Characterization of Ideal Semigroups of Inverse Semigroups
Characterization of Ideal Semigroups of Inverse Semigroups
View Book: https://doi.org/10.9734/bpi/mono/978-81-19491-82-7/CH2
#Ideal_semigroup #fundamental_semigroup #fundamental_inverse_semigroup #munnsemigroup #ideal_congruence
published: 18 Sep 2023
Mark Lawson: "Inverse semigroups and étale groupoids"
Talk given as part of Categories Logic and Physics Scotland: http://homepages.inf.ed.ac.uk/cheunen/clapscotland
I shall describe the adjunction between the category of inverse semigroups and that of étale groupoids and how this leads to a non-commutative version of Stone duality.
published: 10 May 2016
Prof. Adenike Olusola: Structure of Subgroups of Finite Semilattices: Symmetric Inverse Semigroup
Optimization Seminar: Prof. Matthias Koeppe from University of California, Davis
Title: Inverse semigroup theory of cutting planes for integer linear optimiza...
Optimization Seminar: Prof. Matthias Koeppe from University of California, Davis
Title: Inverse semigroup theory of cutting planes for integer linear optimization
Abstract: MIP practitioners can solve large-scale mixed integer optimizationproblems to optimality or near-optimality by competent modelizationand use of branch-and-cut solvers. This technology was enabled to alarge part by the revival of Gomory’s classic general-purpose cuttingplanes such as the Gomory mixed integer cut. In the theory of such general-purpose cutting planes (validinequalities) the traditional, finite-dimensional techniques ofpolyhedral combinatorics are complemented by infinite-dimensionalmethods, the study of cut-generating functions. In my talk I will introduce the classic Gomory-Johnson model, auniversal relaxation of integer programs in the form of a singleconstraint in infinitely many nonnegative integer variables. Thenondominated valid inequalities (cut-generating functions) for thismodel, “minimal functions”, are characterized by functionalinequalities such as subadditivity. Given a minimal function, we are interested in finding improvingdirections that lead to stronger cuts and eventually to “extremefunctions”, which cannot be strengthened further — an analogue offacet-defining inequalities. I will present an inverse semigroup theory for minimal functions,which enables us to obtain a complete description of the space of”improving directions” (perturbations) of a minimal function. This isjoint work with Robert Hildebrand and Yuan Zhou, which appeared inIPCO 2019; a full paper is available at https://arxiv.org/abs/1811.06189 .
Optimization Seminar: Prof. Matthias Koeppe from University of California, Davis
Title: Inverse semigroup theory of cutting planes for integer linear optimization
Abstract: MIP practitioners can solve large-scale mixed integer optimizationproblems to optimality or near-optimality by competent modelizationand use of branch-and-cut solvers. This technology was enabled to alarge part by the revival of Gomory’s classic general-purpose cuttingplanes such as the Gomory mixed integer cut. In the theory of such general-purpose cutting planes (validinequalities) the traditional, finite-dimensional techniques ofpolyhedral combinatorics are complemented by infinite-dimensionalmethods, the study of cut-generating functions. In my talk I will introduce the classic Gomory-Johnson model, auniversal relaxation of integer programs in the form of a singleconstraint in infinitely many nonnegative integer variables. Thenondominated valid inequalities (cut-generating functions) for thismodel, “minimal functions”, are characterized by functionalinequalities such as subadditivity. Given a minimal function, we are interested in finding improvingdirections that lead to stronger cuts and eventually to “extremefunctions”, which cannot be strengthened further — an analogue offacet-defining inequalities. I will present an inverse semigroup theory for minimal functions,which enables us to obtain a complete description of the space of”improving directions” (perturbations) of a minimal function. This isjoint work with Robert Hildebrand and Yuan Zhou, which appeared inIPCO 2019; a full paper is available at https://arxiv.org/abs/1811.06189 .
Optimization Seminar: Prof. Matthias Koeppe from University of California, Davis
Title: Inverse semigroup theory of cutting planes for integer linear optimizat...
Optimization Seminar: Prof. Matthias Koeppe from University of California, Davis
Title: Inverse semigroup theory of cutting planes for integer linear optimization
Abstract: MIP practitioners can solve large-scale mixed integer optimizationproblems to optimality or near-optimality by competent modelizationand use of branch-and-cut solvers. This technology was enabled to alarge part by the revival of Gomory’s classic general-purpose cuttingplanes such as the Gomory mixed integer cut. In the theory of such general-purpose cutting planes (validinequalities) the traditional, finite-dimensional techniques ofpolyhedral combinatorics are complemented by infinite-dimensionalmethods, the study of cut-generating functions. In my talk I will introduce the classic Gomory-Johnson model, auniversal relaxation of integer programs in the form of a singleconstraint in infinitely many nonnegative integer variables. Thenondominated valid inequalities (cut-generating functions) for thismodel, “minimal functions”, are characterized by functionalinequalities such as subadditivity. Given a minimal function, we are interested in finding improvingdirections that lead to stronger cuts and eventually to “extremefunctions”, which cannot be strengthened further — an analogue offacet-defining inequalities. I will present an inverse semigroup theory for minimal functions,which enables us to obtain a complete description of the space of”improving directions” (perturbations) of a minimal function. This isjoint work with Robert Hildebrand and Yuan Zhou, which appeared inIPCO 2019; a full paper is available at https://arxiv.org/abs/1811.06189 .
Optimization Seminar: Prof. Matthias Koeppe from University of California, Davis
Title: Inverse semigroup theory of cutting planes for integer linear optimization
Abstract: MIP practitioners can solve large-scale mixed integer optimizationproblems to optimality or near-optimality by competent modelizationand use of branch-and-cut solvers. This technology was enabled to alarge part by the revival of Gomory’s classic general-purpose cuttingplanes such as the Gomory mixed integer cut. In the theory of such general-purpose cutting planes (validinequalities) the traditional, finite-dimensional techniques ofpolyhedral combinatorics are complemented by infinite-dimensionalmethods, the study of cut-generating functions. In my talk I will introduce the classic Gomory-Johnson model, auniversal relaxation of integer programs in the form of a singleconstraint in infinitely many nonnegative integer variables. Thenondominated valid inequalities (cut-generating functions) for thismodel, “minimal functions”, are characterized by functionalinequalities such as subadditivity. Given a minimal function, we are interested in finding improvingdirections that lead to stronger cuts and eventually to “extremefunctions”, which cannot be strengthened further — an analogue offacet-defining inequalities. I will present an inverse semigroup theory for minimal functions,which enables us to obtain a complete description of the space of”improving directions” (perturbations) of a minimal function. This isjoint work with Robert Hildebrand and Yuan Zhou, which appeared inIPCO 2019; a full paper is available at https://arxiv.org/abs/1811.06189 .
Speaker:
Fernando Lledó, University Carlos III de Madrid and ICMAT Madrid.
Event:
The 48th Canadian Operator Symposium
http://www.fields.utoronto.ca/activi...
Speaker:
Fernando Lledó, University Carlos III de Madrid and ICMAT Madrid.
Event:
The 48th Canadian Operator Symposium
http://www.fields.utoronto.ca/activities/19-20/COSy
Title: Amenability versus paradoxical decompositions for inverse semigroups and their C*-algebras
In this talk I will address the dichotomy between amenability and the existence of paradoxical decompositions for inverse semigroups. Semigroups have singular dynamics in the sense that multiplication by an element can drastically shrink the size of a finite subset in the semigroup. This fact forces to analyze again basic aspects coming from group theory like paradoxical decompositions, Foelner nets or existence of invariant means. We will analyze these aspects in relation to two C*-algebras that generalize the usual reduced group C*-algebra as well as the uniform Roe algebra of a countable finitely generated group.
Speaker:
Fernando Lledó, University Carlos III de Madrid and ICMAT Madrid.
Event:
The 48th Canadian Operator Symposium
http://www.fields.utoronto.ca/activities/19-20/COSy
Title: Amenability versus paradoxical decompositions for inverse semigroups and their C*-algebras
In this talk I will address the dichotomy between amenability and the existence of paradoxical decompositions for inverse semigroups. Semigroups have singular dynamics in the sense that multiplication by an element can drastically shrink the size of a finite subset in the semigroup. This fact forces to analyze again basic aspects coming from group theory like paradoxical decompositions, Foelner nets or existence of invariant means. We will analyze these aspects in relation to two C*-algebras that generalize the usual reduced group C*-algebra as well as the uniform Roe algebra of a countable finitely generated group.
Talk given as part of Categories Logic and Physics Scotland: http://homepages.inf.ed.ac.uk/cheunen/clapscotland
I shall describe the adjunction between the ca...
Talk given as part of Categories Logic and Physics Scotland: http://homepages.inf.ed.ac.uk/cheunen/clapscotland
I shall describe the adjunction between the category of inverse semigroups and that of étale groupoids and how this leads to a non-commutative version of Stone duality.
Talk given as part of Categories Logic and Physics Scotland: http://homepages.inf.ed.ac.uk/cheunen/clapscotland
I shall describe the adjunction between the category of inverse semigroups and that of étale groupoids and how this leads to a non-commutative version of Stone duality.
Optimization Seminar: Prof. Matthias Koeppe from University of California, Davis
Title: Inverse semigroup theory of cutting planes for integer linear optimization
Abstract: MIP practitioners can solve large-scale mixed integer optimizationproblems to optimality or near-optimality by competent modelizationand use of branch-and-cut solvers. This technology was enabled to alarge part by the revival of Gomory’s classic general-purpose cuttingplanes such as the Gomory mixed integer cut. In the theory of such general-purpose cutting planes (validinequalities) the traditional, finite-dimensional techniques ofpolyhedral combinatorics are complemented by infinite-dimensionalmethods, the study of cut-generating functions. In my talk I will introduce the classic Gomory-Johnson model, auniversal relaxation of integer programs in the form of a singleconstraint in infinitely many nonnegative integer variables. Thenondominated valid inequalities (cut-generating functions) for thismodel, “minimal functions”, are characterized by functionalinequalities such as subadditivity. Given a minimal function, we are interested in finding improvingdirections that lead to stronger cuts and eventually to “extremefunctions”, which cannot be strengthened further — an analogue offacet-defining inequalities. I will present an inverse semigroup theory for minimal functions,which enables us to obtain a complete description of the space of”improving directions” (perturbations) of a minimal function. This isjoint work with Robert Hildebrand and Yuan Zhou, which appeared inIPCO 2019; a full paper is available at https://arxiv.org/abs/1811.06189 .
Optimization Seminar: Prof. Matthias Koeppe from University of California, Davis
Title: Inverse semigroup theory of cutting planes for integer linear optimization
Abstract: MIP practitioners can solve large-scale mixed integer optimizationproblems to optimality or near-optimality by competent modelizationand use of branch-and-cut solvers. This technology was enabled to alarge part by the revival of Gomory’s classic general-purpose cuttingplanes such as the Gomory mixed integer cut. In the theory of such general-purpose cutting planes (validinequalities) the traditional, finite-dimensional techniques ofpolyhedral combinatorics are complemented by infinite-dimensionalmethods, the study of cut-generating functions. In my talk I will introduce the classic Gomory-Johnson model, auniversal relaxation of integer programs in the form of a singleconstraint in infinitely many nonnegative integer variables. Thenondominated valid inequalities (cut-generating functions) for thismodel, “minimal functions”, are characterized by functionalinequalities such as subadditivity. Given a minimal function, we are interested in finding improvingdirections that lead to stronger cuts and eventually to “extremefunctions”, which cannot be strengthened further — an analogue offacet-defining inequalities. I will present an inverse semigroup theory for minimal functions,which enables us to obtain a complete description of the space of”improving directions” (perturbations) of a minimal function. This isjoint work with Robert Hildebrand and Yuan Zhou, which appeared inIPCO 2019; a full paper is available at https://arxiv.org/abs/1811.06189 .
Speaker:
Fernando Lledó, University Carlos III de Madrid and ICMAT Madrid.
Event:
The 48th Canadian Operator Symposium
http://www.fields.utoronto.ca/activities/19-20/COSy
Title: Amenability versus paradoxical decompositions for inverse semigroups and their C*-algebras
In this talk I will address the dichotomy between amenability and the existence of paradoxical decompositions for inverse semigroups. Semigroups have singular dynamics in the sense that multiplication by an element can drastically shrink the size of a finite subset in the semigroup. This fact forces to analyze again basic aspects coming from group theory like paradoxical decompositions, Foelner nets or existence of invariant means. We will analyze these aspects in relation to two C*-algebras that generalize the usual reduced group C*-algebra as well as the uniform Roe algebra of a countable finitely generated group.
Talk given as part of Categories Logic and Physics Scotland: http://homepages.inf.ed.ac.uk/cheunen/clapscotland
I shall describe the adjunction between the category of inverse semigroups and that of étale groupoids and how this leads to a non-commutative version of Stone duality.
In mathematics, an inverse semigroup (occasionally called an inversion semigroup) S is a semigroup in which every element x in S has a unique inversey in S in the sense that x = xyx and y = yxy, i.e. a regular semigroup in which every element has a unique inverse. Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of partial symmetries.
(The convention followed in this article will be that of writing a function on the right of its argument, and
composing functions from left to right — a convention often observed in semigroup theory.)