The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left...The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left weak H-module coalgebra.First,a weak generalized smash coproduct C×lH D over quantum groupoids is defined and the module and comodule structures on it are constructed.The weak generalized right smash coproduct C×rL D is similar.Then some isomorph-isms between them are obtained.Secondly,by introducing some concepts of a weak convolution invertible element,a weak co-inner coaction and a strongly relative co-inner coaction,a sufficient condition for C×rH D to be isomorphic to Cv D is obtained,where v∈WC（C,H）and the coaction of H on D is right strongly relative co-inner.Finally,the duality theorem for a generalized smash coproduct over quantum groupoids,（C×lH H）×lH H≌Cv（H×lH H）,is obtained.展开更多
After introducing some of the basic definitions and results from the theory of groupoid and Lie algebroid,we investigate the discrete Lagrangian mechanics from the viewpoint of groupoid theory and give the connection ...After introducing some of the basic definitions and results from the theory of groupoid and Lie algebroid,we investigate the discrete Lagrangian mechanics from the viewpoint of groupoid theory and give the connection betweengroupoids variation and the methods of the first and second discrete variational principles.展开更多
First,we prove a decomposition formula for any multiplicative differential form on a Lie groupoid G.Next,we prove that if G is a Poisson Lie groupoid,then the spaceΩ_(mult)·(G)of multiplicative forms on G has a ...First,we prove a decomposition formula for any multiplicative differential form on a Lie groupoid G.Next,we prove that if G is a Poisson Lie groupoid,then the spaceΩ_(mult)·(G)of multiplicative forms on G has a differential graded Lie algebra(DGLA)structure.Furthermore,when combined withΩ·(M),which is the space of forms on the base manifold M of G,Ω_(mult)·(G)forms a canonical DGLA crossed module.This supplements a previously known fact that multiplicative multi-vector fields on G form a DGLA crossed module with the Schouten algebraΓ(∧·A)stemming from the Lie algebroid A of G.展开更多
We show that every source connected Lie groupoid always has global bisections through any given point. This bisection can be chosen to be the multiplication of some exponentials as close as possible to a prescribed cu...We show that every source connected Lie groupoid always has global bisections through any given point. This bisection can be chosen to be the multiplication of some exponentials as close as possible to a prescribed curve. The existence of bisections through more than one prescribed point is also discussed. We give some interesting applications of these results.展开更多
A notion of an irreducible representation, as well as of a square integrable representation on an arbitrary locally compact groupoid, is introduced. A generalization of a version of Schur＇s lemma on a locally compact...A notion of an irreducible representation, as well as of a square integrable representation on an arbitrary locally compact groupoid, is introduced. A generalization of a version of Schur＇s lemma on a locally compact groupoid is given. This is used in order to extend some well-known results from locally compact groups to the case of locally compact groupoids. Indeed, we have proved that if L is a continuous irreducible representation of a compact groupoid G defined by a continuous Hilbert bundle H = （Hu）u∈G^0, then each Hu is finite dimensional. It is also shown that if L is an irreducible representation of a principal locally compact groupoid defined by a Hilbert bundle （G^0, （Hu）,μ）, then dimHu = 1 （u ∈ G^0）. Furthermore it is proved that every square integrable representation of a locally compact groupoid is unitary equivalent to a subrepresentation of the left regular representation. Furthermore, for r-discrete groupoids, it is shown that every irreducible subrepresentation of the left regular representation is square integrable.展开更多
Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bound...Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bounded functions L<sup>∞</sup>(X, μ) on X. We confirm that the commutative von Neumann algebras M⊂B(H), with H=L<sup>2</sup>(X, μ), are unitary equivariant to the maximal ideals of the commutative algebra C(X). Subsequenly, we use the measure groupoid to formulate the algebraic and topological structures of the commutative algebra C(X) following its action on M(X) and define its representation and ergodic dynamical system on the commutative von Neumann algebras of M of B(H) .展开更多
Proposed here is a new framework for the analysis of complex systems as a non-explicitly programmed mathematical hierarchy of subsystems using only the fundamental principle of causality, the mathematics of groupoid s...Proposed here is a new framework for the analysis of complex systems as a non-explicitly programmed mathematical hierarchy of subsystems using only the fundamental principle of causality, the mathematics of groupoid symmetries, and a basic causal metric needed to support measurement in Physics. The complex system is described as a discrete set S of state variables. Causality is described by an acyclic partial order w on S, and is considered as a constraint on the set of allowed state transitions. Causal set (S, w) is the mathematical model of the system. The dynamics it describes is uncertain. Consequently, we focus on invariants, particularly group-theoretical block systems. The symmetry of S by itself is characterized by its symmetric group, which generates a trivial block system over S. The constraint of causality breaks this symmetry and degrades it to that of a groupoid, which may yield a non-trivial block system on S. In addition, partial order w determines a partial order for the blocks, and the set of blocks becomes a causal set with its own, smaller block system. Recursion yields a multilevel hierarchy of invariant blocks over S with the properties of a scale-free mathematical fractal. This is the invariant being sought. The finding hints at a deep connection between the principle of causality and a class of poorly understood phenomena characterized by the formation of hierarchies of patterns, such as emergence, selforganization, adaptation, intelligence, and semantics. The theory and a thought experiment are discussed and previous evidence is referenced. Several predictions in the human brain are confirmed with wide experimental bases. Applications are anticipated in many disciplines, including Biology, Neuroscience, Computation, Artificial Intelligence, and areas of Engineering such as system autonomy, robotics, systems integration, and image and voice recognition.展开更多
The big problem of Big Data is the lack of a machine learning process that scales and finds meaningful features. Humans fill in for the insufficient automation, but the complexity of the tasks outpaces the human mind...The big problem of Big Data is the lack of a machine learning process that scales and finds meaningful features. Humans fill in for the insufficient automation, but the complexity of the tasks outpaces the human mind’s capacity to comprehend the data. Heuristic partition methods may help but still need humans to adjust the parameters. The same problems exist in many other disciplines and technologies that depend on Big Data or Machine Learning. Proposed here is a fractal groupoid-theoretical method that recursively partitions the problem and requires no heuristics or human intervention. It takes two steps. First, make explicit the fundamental causal nature of information in the physical world by encoding it as a causal set. Second, construct a functor F: C C′ on the category of causal sets that morphs causal set C into smaller causal set C′ by partitioning C into a set of invariant groupoid-theoretical blocks. Repeating the construction, there arises a sequence of progressively smaller causal sets C, C′, C″, … The sequence defines a fractal hierarchy of features, with the features being invariant and hence endowed with a physical meaning, and the hierarchy being scale-free and hence ensuring proper scaling at all granularities. Fractals exist in nature nearly everywhere and at all physical scales, and invariants have long been known to be meaningful to us. The theory is also of interest for NP-hard combinatorial problems that can be expressed as a causal set, such as the Traveling Salesman problem. The recursive groupoid partition promoted by functor F works against their combinatorial complexity and appears to allow a low-order polynomial solution. A true test of this property requires special hardware, not yet available. However, as a proof of concept, a suite of sequential, non-heuristic algorithms were developed and used to solve a real-world 120-city problem of TSP on a personal computer. The results are reported.展开更多
We introduce the notions of implicative ideals and fuzzy implicative ideals of a distributive implication groupoid. Some properties of these ideals will be investigated. In particular, the necessary and sufficient con...We introduce the notions of implicative ideals and fuzzy implicative ideals of a distributive implication groupoid. Some properties of these ideals will be investigated. In particular, the necessary and sufficient conditions for an ideal（fuzzy ideal） to be an implicative ideal（fuzzy implicative ideal） is given. By using the concept of level sets, we will characterize the fuzzy implicative ideals of a distributive implication groupoid. Finally, an extension property for fuzzy implicative ideals is given.展开更多
In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern language, their idea has been ...In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern language, their idea has been to use the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the nonlinear Spencer sequence for the Lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of electromagnetism (EM), with the only experimental need to measure the EM constant in vacuum. With a manifold of dimension n, the difficulty is to deal with the n nonlinear transformations that have been called “elations” by E. Cartan in 1922. Using the fact that dimension n=4 has very specific properties for the computation of the Spencer cohomology, we prove that there is thus no conceptual difference between the Cosserat EL field or induction equations and the Maxwell EM field or induction equations. As a byproduct, the well known field/matter couplings (piezzoelectricity, photoelasticity, streaming birefringence, …) can be described abstractly, with the only experimental need to measure the corresponding coupling constants. The main consequence of this paper is the need to revisit the mathematical foundations of gauge theory (GT) because we have proved that EM was depending on the conformal group and not on U(1), with a shift by one step to the left in the physical interpretation of the differential sequence involved.展开更多
In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern framework, they used the nonli...In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern framework, they used the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the linear and nonlinear Spencer sequences for the Lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of both electromagnetism (EM) and gravitation (GR), with the only experimental need to measure the EM and GR constants. With a manifold of dimension n ≥ 3, the difficulty is to deal with the n nonlinear transformations that have been called “elations” by E. Cartan in 1922. Using the fact that dimension n = 4 has very specific properties for the computation of the Spencer cohomology, we also prove that there is no conceptual difference between the (nonlinear) Cosserat EL field or induction equations and the (linear) Maxwell EM field or induction equations. As for gravitation, the dimension n = 4 also allows to have a conformal factor defined everywhere but at the central attractive mass because the inversion law of the isotropy subgroupoid made by second order jets transforms attraction into repulsion. The mathematical foundations of both electromagnetism and gravitation are thus only depending on the structure of the conformal pseudogroup of space-time.展开更多
The internal Zappa-Szép products emerge when a semigroup has the property that every element has a unique decomposition as a product of elements from two given subsemigroups. The external version constructed from...The internal Zappa-Szép products emerge when a semigroup has the property that every element has a unique decomposition as a product of elements from two given subsemigroups. The external version constructed from actions of two semigroups on one another satisfying axiom derived by G. Zappa. We illustrate the correspondence between the two versions internal and the external of Zappa-Szép products of semigroups. We consider the structure of the internal Zappa-Szép product as an enlargement. We show how rectangular band can be described as the Zappa-Szép product of a left-zero semigroup and a right-zero semigroup. We find necessary and sufficient conditions for the Zappa-Szép product of regular semigroups to again be regular, and necessary conditions for the Zappa-Szép product of inverse semigroups to again be inverse. We generalize the Billhardt λ-semidirect product to the Zappa-Szép product of a semilattice E and a group G by constructing an inductive groupoid.展开更多
In this paper, we define the notions of orbifold loop and orbifold road, with which, we reformulate the definition of orbifold fundamental group and deck translation group. We show that, for each orbifold, the orbifol...In this paper, we define the notions of orbifold loop and orbifold road, with which, we reformulate the definition of orbifold fundamental group and deck translation group. We show that, for each orbifold, the orbifold fundamental group is isomorphic to the deck translation group.展开更多
We introduce notions of continuous orbit equivalence and its one-sided version for countable left Ore semigroup actions on compact spaces by surjective local homeomorphisms,and characterize them in terms of the corres...We introduce notions of continuous orbit equivalence and its one-sided version for countable left Ore semigroup actions on compact spaces by surjective local homeomorphisms,and characterize them in terms of the corresponding transformation groupoids and their operator algebras.In particular,we show that two essentially free semigroup actions on totally disconnected compact spaces are continuously orbit equivalent if and only if there is a canonical abelian subalgebra preserving C^(∗)-isomorphism between the associated transformation groupoid C^(∗)-algebras.We also give some examples of orbit equivalence,consider the special case of semigroup actions by homeomorphisms and relate continuous orbit equivalence of semigroup actions to that of the associated group actions.展开更多
We introduce the relations Lu and Ru with respect to a subset U of idempotents. Based on Lv and Rv, we define a new class of semigroups which we name U-concordant semigroups. Our purpose is to describe U-concordant se...We introduce the relations Lu and Ru with respect to a subset U of idempotents. Based on Lv and Rv, we define a new class of semigroups which we name U-concordant semigroups. Our purpose is to describe U-concordant semigroups by generalized categories over a regular biordered set. We show that the category of U-concordant semigroups and admissible morphisms is isomorphic to the category of RBS generalized categories and pseudo functors. Our approach is inspired from Armstrong＇s work on the connection between regular biordered sets and concordant semigroups. The significant difference in strategy is by using RBS generalized categories equipped with pre-orders, we have no need to discuss the quotient of a category factored by a congruence.展开更多
基金The National Natural Science Foundation of China(No.10871042)the Natural Science Foundation of Jiangsu Province(No.BK2009258)
文摘The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left weak H-module coalgebra.First,a weak generalized smash coproduct C×lH D over quantum groupoids is defined and the module and comodule structures on it are constructed.The weak generalized right smash coproduct C×rL D is similar.Then some isomorph-isms between them are obtained.Secondly,by introducing some concepts of a weak convolution invertible element,a weak co-inner coaction and a strongly relative co-inner coaction,a sufficient condition for C×rH D to be isomorphic to Cv D is obtained,where v∈WC（C,H）and the coaction of H on D is right strongly relative co-inner.Finally,the duality theorem for a generalized smash coproduct over quantum groupoids,（C×lH H）×lH H≌Cv（H×lH H）,is obtained.
基金National Key Basic Research Project of China under Grant No.2004CB318000National Natural Science Foundation of China under Grant Nos.10375038 and 90403018
文摘After introducing some of the basic definitions and results from the theory of groupoid and Lie algebroid,we investigate the discrete Lagrangian mechanics from the viewpoint of groupoid theory and give the connection betweengroupoids variation and the methods of the first and second discrete variational principles.
基金supported by National Natural Science Foundation of China(Grant No.12071241)the National Key Research and Development Program of China(Grant No.2021YFA1002000).
文摘First,we prove a decomposition formula for any multiplicative differential form on a Lie groupoid G.Next,we prove that if G is a Poisson Lie groupoid,then the spaceΩ_(mult)·(G)of multiplicative forms on G has a differential graded Lie algebra(DGLA)structure.Furthermore,when combined withΩ·(M),which is the space of forms on the base manifold M of G,Ω_(mult)·(G)forms a canonical DGLA crossed module.This supplements a previously known fact that multiplicative multi-vector fields on G form a DGLA crossed module with the Schouten algebraΓ(∧·A)stemming from the Lie algebroid A of G.
文摘We show that every source connected Lie groupoid always has global bisections through any given point. This bisection can be chosen to be the multiplication of some exponentials as close as possible to a prescribed curve. The existence of bisections through more than one prescribed point is also discussed. We give some interesting applications of these results.
基金Supported by the office of Graduate Studies and the Center of Excellence for Mathematics of the University of Isfahan
文摘A notion of an irreducible representation, as well as of a square integrable representation on an arbitrary locally compact groupoid, is introduced. A generalization of a version of Schur＇s lemma on a locally compact groupoid is given. This is used in order to extend some well-known results from locally compact groups to the case of locally compact groupoids. Indeed, we have proved that if L is a continuous irreducible representation of a compact groupoid G defined by a continuous Hilbert bundle H = （Hu）u∈G^0, then each Hu is finite dimensional. It is also shown that if L is an irreducible representation of a principal locally compact groupoid defined by a Hilbert bundle （G^0, （Hu）,μ）, then dimHu = 1 （u ∈ G^0）. Furthermore it is proved that every square integrable representation of a locally compact groupoid is unitary equivalent to a subrepresentation of the left regular representation. Furthermore, for r-discrete groupoids, it is shown that every irreducible subrepresentation of the left regular representation is square integrable.
文摘Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bounded functions L<sup>∞</sup>(X, μ) on X. We confirm that the commutative von Neumann algebras M⊂B(H), with H=L<sup>2</sup>(X, μ), are unitary equivariant to the maximal ideals of the commutative algebra C(X). Subsequenly, we use the measure groupoid to formulate the algebraic and topological structures of the commutative algebra C(X) following its action on M(X) and define its representation and ergodic dynamical system on the commutative von Neumann algebras of M of B(H) .
文摘Proposed here is a new framework for the analysis of complex systems as a non-explicitly programmed mathematical hierarchy of subsystems using only the fundamental principle of causality, the mathematics of groupoid symmetries, and a basic causal metric needed to support measurement in Physics. The complex system is described as a discrete set S of state variables. Causality is described by an acyclic partial order w on S, and is considered as a constraint on the set of allowed state transitions. Causal set (S, w) is the mathematical model of the system. The dynamics it describes is uncertain. Consequently, we focus on invariants, particularly group-theoretical block systems. The symmetry of S by itself is characterized by its symmetric group, which generates a trivial block system over S. The constraint of causality breaks this symmetry and degrades it to that of a groupoid, which may yield a non-trivial block system on S. In addition, partial order w determines a partial order for the blocks, and the set of blocks becomes a causal set with its own, smaller block system. Recursion yields a multilevel hierarchy of invariant blocks over S with the properties of a scale-free mathematical fractal. This is the invariant being sought. The finding hints at a deep connection between the principle of causality and a class of poorly understood phenomena characterized by the formation of hierarchies of patterns, such as emergence, selforganization, adaptation, intelligence, and semantics. The theory and a thought experiment are discussed and previous evidence is referenced. Several predictions in the human brain are confirmed with wide experimental bases. Applications are anticipated in many disciplines, including Biology, Neuroscience, Computation, Artificial Intelligence, and areas of Engineering such as system autonomy, robotics, systems integration, and image and voice recognition.
文摘The big problem of Big Data is the lack of a machine learning process that scales and finds meaningful features. Humans fill in for the insufficient automation, but the complexity of the tasks outpaces the human mind’s capacity to comprehend the data. Heuristic partition methods may help but still need humans to adjust the parameters. The same problems exist in many other disciplines and technologies that depend on Big Data or Machine Learning. Proposed here is a fractal groupoid-theoretical method that recursively partitions the problem and requires no heuristics or human intervention. It takes two steps. First, make explicit the fundamental causal nature of information in the physical world by encoding it as a causal set. Second, construct a functor F: C C′ on the category of causal sets that morphs causal set C into smaller causal set C′ by partitioning C into a set of invariant groupoid-theoretical blocks. Repeating the construction, there arises a sequence of progressively smaller causal sets C, C′, C″, … The sequence defines a fractal hierarchy of features, with the features being invariant and hence endowed with a physical meaning, and the hierarchy being scale-free and hence ensuring proper scaling at all granularities. Fractals exist in nature nearly everywhere and at all physical scales, and invariants have long been known to be meaningful to us. The theory is also of interest for NP-hard combinatorial problems that can be expressed as a causal set, such as the Traveling Salesman problem. The recursive groupoid partition promoted by functor F works against their combinatorial complexity and appears to allow a low-order polynomial solution. A true test of this property requires special hardware, not yet available. However, as a proof of concept, a suite of sequential, non-heuristic algorithms were developed and used to solve a real-world 120-city problem of TSP on a personal computer. The results are reported.
文摘We introduce the notions of implicative ideals and fuzzy implicative ideals of a distributive implication groupoid. Some properties of these ideals will be investigated. In particular, the necessary and sufficient conditions for an ideal（fuzzy ideal） to be an implicative ideal（fuzzy implicative ideal） is given. By using the concept of level sets, we will characterize the fuzzy implicative ideals of a distributive implication groupoid. Finally, an extension property for fuzzy implicative ideals is given.
文摘In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern language, their idea has been to use the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the nonlinear Spencer sequence for the Lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of electromagnetism (EM), with the only experimental need to measure the EM constant in vacuum. With a manifold of dimension n, the difficulty is to deal with the n nonlinear transformations that have been called “elations” by E. Cartan in 1922. Using the fact that dimension n=4 has very specific properties for the computation of the Spencer cohomology, we prove that there is thus no conceptual difference between the Cosserat EL field or induction equations and the Maxwell EM field or induction equations. As a byproduct, the well known field/matter couplings (piezzoelectricity, photoelasticity, streaming birefringence, …) can be described abstractly, with the only experimental need to measure the corresponding coupling constants. The main consequence of this paper is the need to revisit the mathematical foundations of gauge theory (GT) because we have proved that EM was depending on the conformal group and not on U(1), with a shift by one step to the left in the physical interpretation of the differential sequence involved.
文摘In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern framework, they used the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the linear and nonlinear Spencer sequences for the Lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of both electromagnetism (EM) and gravitation (GR), with the only experimental need to measure the EM and GR constants. With a manifold of dimension n ≥ 3, the difficulty is to deal with the n nonlinear transformations that have been called “elations” by E. Cartan in 1922. Using the fact that dimension n = 4 has very specific properties for the computation of the Spencer cohomology, we also prove that there is no conceptual difference between the (nonlinear) Cosserat EL field or induction equations and the (linear) Maxwell EM field or induction equations. As for gravitation, the dimension n = 4 also allows to have a conformal factor defined everywhere but at the central attractive mass because the inversion law of the isotropy subgroupoid made by second order jets transforms attraction into repulsion. The mathematical foundations of both electromagnetism and gravitation are thus only depending on the structure of the conformal pseudogroup of space-time.
文摘The internal Zappa-Szép products emerge when a semigroup has the property that every element has a unique decomposition as a product of elements from two given subsemigroups. The external version constructed from actions of two semigroups on one another satisfying axiom derived by G. Zappa. We illustrate the correspondence between the two versions internal and the external of Zappa-Szép products of semigroups. We consider the structure of the internal Zappa-Szép product as an enlargement. We show how rectangular band can be described as the Zappa-Szép product of a left-zero semigroup and a right-zero semigroup. We find necessary and sufficient conditions for the Zappa-Szép product of regular semigroups to again be regular, and necessary conditions for the Zappa-Szép product of inverse semigroups to again be inverse. We generalize the Billhardt λ-semidirect product to the Zappa-Szép product of a semilattice E and a group G by constructing an inductive groupoid.
基金Supported by General Project of Science Research of Guangzhou(Grant No.2017070126)National Natural Science Foundation of China(Grant No.11226034).
文摘In this paper, we define the notions of orbifold loop and orbifold road, with which, we reformulate the definition of orbifold fundamental group and deck translation group. We show that, for each orbifold, the orbifold fundamental group is isomorphic to the deck translation group.
基金Supported by the NSF of China(Grant No.12271469,11771379,11971419)。
文摘We introduce notions of continuous orbit equivalence and its one-sided version for countable left Ore semigroup actions on compact spaces by surjective local homeomorphisms,and characterize them in terms of the corresponding transformation groupoids and their operator algebras.In particular,we show that two essentially free semigroup actions on totally disconnected compact spaces are continuously orbit equivalent if and only if there is a canonical abelian subalgebra preserving C^(∗)-isomorphism between the associated transformation groupoid C^(∗)-algebras.We also give some examples of orbit equivalence,consider the special case of semigroup actions by homeomorphisms and relate continuous orbit equivalence of semigroup actions to that of the associated group actions.
基金This research was supported by the NSFC （Grant No. 11471255, 11501331）. The second author was supported by the Shandong Province Natural Science Foundation （Grant No. BS2015SF002）, SDUST Research Fund （No. 2014TDJH102）, and Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province.
文摘We introduce the relations Lu and Ru with respect to a subset U of idempotents. Based on Lv and Rv, we define a new class of semigroups which we name U-concordant semigroups. Our purpose is to describe U-concordant semigroups by generalized categories over a regular biordered set. We show that the category of U-concordant semigroups and admissible morphisms is isomorphic to the category of RBS generalized categories and pseudo functors. Our approach is inspired from Armstrong＇s work on the connection between regular biordered sets and concordant semigroups. The significant difference in strategy is by using RBS generalized categories equipped with pre-orders, we have no need to discuss the quotient of a category factored by a congruence.
