Some relations on completely regular semigroups

A semigroup S is called a Clifford semigroup if it is completely regular and inverse. In this paper, some relations related to the least Clifford semigroup congruences on completely regular semigroups are characterized. We give the relation between Y and ξ on completely regular semigroups and get that Y ^* is contained in the least Clifford congruence on completely regular semigroups generally. Further, we consider the relation Y ^*, Y, ν and ε on completely simple semigroups and completely regular semigroups. This work is supported by Leading Academic Discipline Project of Shanghai Normal University, Project Number: DZL803 and General Scientific Research Project of Shanghai Normal University, No. SK200707.     

Topological realizations of families of ergodic automorphisms, multitowers and orbit equivalence

We study minimal topological realizations of families of ergodic measure preserving automorphisms (e.m.p.a.'s). Our main result is the following theorem. Theorem: Let {T_p:p∈I} be an arbitrary finite or countable collection of e.m.p.a.'s on nonatomic Lebesgue probability spaces (Y _p v _p ). Let S be a Cantor minimal system such that the cardinality of the set ε _S of all ergodic S-invariant Borel probability measures is at least the cardinality of I. Then for any collection {μ _p :pεI} of distinct measures from ε _S there is a Cantor minimal system S′ in the topological orbit equivalence class of S such that, as a measure preserving system, (S ¹,μ _p ) is isomorphic to T_p for every p∈I. Moreover, S′ can be chosen strongly orbit equivalent to S if and only if all finite topological factors of S are measure-theoretic factors of T_p for all p∈I. This result shows, in particular, that there are no restrictions at all for the topological realizations of countable families of e.m.p.a.'s in Cantor minimal systems. Namely, for any finite or countable collection {T ₁,T₂,…} of e.m.p.a.'s of nonatomic Lebesgue probability spaces, there is a Cantor minimal systemS, whose collection {μ1,μ2…} of ergodic Borel probability measures is in one-to-one correspondence with {T ₁,T₂,…}, and such that (S,μ _i ) is isomorphic toT _i for alli. Furthermore, since realizations are taking place within orbit equivalence classes of a given Cantor minimal system, our results generalize the strong orbit realization theorem and the orbit realization theorem of [18]. Those theorems are now special cases of our result where the collections {T _p}, {T _p }{μ _p } consist of just one element each. Research of I.K. was supported by NSF grant DMS 0140068.     

P-systems in regular semigroups

In this paper, firstly it is shown that a regular semigroup S becomes a regular *-semigroup (in the sense of [1]) if and only if S has a certain subset called a p-system. Secondly, all the normal *-bands are completely described in terms of rectangular *-bands (square bands) and transitive systems of homomorphisms of rectangular *-bands. Further, it is shown that an orthodox semigroup S becomes a regular *-semigroup if there is a p-system F of the band E_S of idempotents of S such that F∋e, E_S∋t, e≥t imply t∈F. By using this result, it is also shown that F is a p-system of a generalized inverse semigroup S if and only if F is a p-system of F_S. Dedicated to Professor L. M. Gluskin on his 60th birthday     

The structure of bisimple left unipotent semigroups as ordered pairs

Call a semigroup S left unipotent if eachℒ-class of S contains exactly one idempotent. A structure theorem for bisimple left unipotent semigroups is given which reduces to that of N. R. Reilly [8] for bisimple inverse semigroups. A structure theorem, alternative to one given by R. J. Warne [13], is given for the case when the band E_S of idempotents of S is an ω-chain of right zero semigroups, and two applications of it are made. This research was partially supported by a grant from the National Science Foundation.     

Morita Equivalence for Factorisable Semigroups

Recall that the semigroups S and R are said to be strongly Morita equivalent if there exists a unitary Morita context (S, R., _S P _R,R Q _S ,〈〉 , ⌈⌉) with 〈〉 and ⌈⌉ surjective. For a factorisable semigroup S, we denote ζ _S = {(s ₁, s ₂) ∈S×S|ss ₁ = ss ₂, ∀s∈S}, S' = S/ζ _S and US-FAct = { _S M∈S− Act |SM = M and SHom _S (S, M) ≅M}. We show that, for factorisable semigroups S and M, the categories US-FAct and UR-FAct are equivalent if and only if the semigroups S' and R' are strongly Morita equivalent. Some conditions for a factorisable semigroups to be strongly Morita equivalent to a sandwich semigroup, local units semigroup, monoid and group separately are also given. Moreover, we show that a semigroup S is completely simple if and only if S is strongly Morita equivalent to a group and for any index set I, S⊗SHom _S (S, ∐ _i∈I S) →∐ _i∈I S, s⊗t·ƒ↦ (st)ƒ is an S-isomorphism. The research is partially supported by a UGC(HK) grant #2160092. Project is supported by the National Natural Science Foundation of China     

Mixed weak continuity on generalized topological spaces

We introduce the notion of mixed weak (μ,ν₁ν₂)-continuity between a generalized topology μ and two generalized topologies ν₁, ν₂. We characterize such continuity in terms of mixed generalized open sets: (ν₁,ν₂)′-semiopen sets, (ν₁,ν₂)′-preopen sets, (ν₁,ν₂)-preopen sets [2], (ν₁,ν₂)′-β′-open sets and θ(ν₁,ν₂)-open sets [3]. In particular, we show that for a given mixed weakly (μ,ν₁ν₂)-continuous function, if the codomain of the given function is mixed regular (=(ν₁,ν₂)-regular), then the function is also (μ,ν₁)-continuous.     

Zur Theorie der Stetigen Teilbarkeitshalbgruppen

O. CallS:=(S,·,∩) a d-semigroup ifS satisfies the axioms (A1) (S,·) is a semigroup, (A2) (S,∩) is a semilattice (A3), (S,·,∩) is a semiring, (A4) a ≤b⇒bε aS ∩ Sa. Call tεS positive if ÅaεS: ta ≥a≤at. Let S⁺ denote the set {t‖t positive}. Every d-semigroup is closed under sup and (s,·,∪) is a semiring, (S, ∩, ∪) is a distributive lattice. Denote by D□X the implication s=Xa_i⇒x□s□y=X(x□a_i□y) where □ε{·,∩,∪} and Xε{∪,∩}. CallS continuous ifS satisfies all D□X. The theory of d-semigroups (divisibility-semigroups) was established in [3], [4], [5], and is continued here by some contributions to the theory of continuous d-semigroups the main results of which are the two propositions: (1) LetS be a d-semigroup with 1. ThenS satisfies D□X iffS ⁺ satisfies this axiom. (2) LetS be continuous. Then (S,·) is commutative. Obviously Proposition (2) is an improvement of Iwasawa's theorem concerning conditionally complete lattice ordered groups.

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Zur Theorie der Stetigen Teilbarkeitshalbgruppen

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Klaus Wagner zum 70. Geburtstag gewidmet     

O-simple strictly regular semigroups

In the previous paper [6], it has been proved that a semigroup S is strictly regular if and only if S is isomorphic to a quasi-direct product EX Λ of a band E and an inverse semigroup Λ. The main purpose of this paper is to present the following results and some relevant matters: (1) A quasi-direct product EX Λ of a band E and an inverse semigroup Λ is simple [bisimple] if and only if Λ is simple [bisimple], and (2) in case where EX Λ has a zero element, EX Λ is O-simple [O-bisimple] if and only if Λ is O-simple [O-bisimple]. Any notation and terminology should be referred to [1], [5] and [6], unless otherwise stated.     

Representability of Hom implies flatness

LetX be a projective scheme over a noetherian base schemeS, and letF be a coherent sheaf onX. For any coherent sheaf ε onX, consider the set-valued contravariant functor Hom(ε,F)S-schemes, defined by Hom(ε,F) (T)= Hom(ε _T ,F _T) where ε _T andF _T are the pull-backs of ε andF toX _T =X x _S T. A basic result of Grothendieck ([EGA], III 7.7.8, 7.7.9) says that ifF is flat over S then Kom_ε,F) is representable for all ε. We prove the converse of the above, in fact, we show that ifL is a relatively ample line bundle onX over S such that the functor Hom(L ^-n ,F) is representable for infinitely many positive integersn, thenF is flat overS. As a corollary, takingX =S, it follows that ifF is a coherent sheaf on S then the functorT ↦H°(T, F _t) on the category ofS-schemes is representable if and only ifF is locally free onS. This answers a question posed by Angelo Vistoli. The techniques we use involve the proof of flattening stratification, together with the methods used in proving the author’s earlier result (see [N1]) that the automorphism group functor of a coherent sheaf onS is representable if and only if the sheaf is locally free.     

Testing linear operators—An average case study

Given a specification linear operatorS, we want to test an implementation linear operatorA and determine whether it conforms to the specification operator according to an error criterion. In an earlier paper [3], we studied a worst case error in which we test whether the error is no more than a given bound ε>0 for all elements in a given setF, i.e., sup _{fεf∥Sf—Af∥≤ε}. In this work, we study the average error instead, i. e., ∫ _F ∥Sf-Af∥²μ(df)ɛ≤², where μ is a probability measure onF. We assume that an upper boundK on the norm of the difference ofS andA is given a priori. It turns out that any finite number of tests is in general inconclusive with the average error. Therefore, as in the worst case, we allow a relaxation parameter α>0 and test for weak conformance with an error bound (1+α)ε. Then a finite number of tests from an arbitrary orthogonal complete sequence is conclusive. Furthermore, the eigenvectors of the covariance operatorC _μ of the probability measure μ provide an almost optimal test sequence. This implies that the test set isuniversal; it only depends on the set of valid inputsF and the measure μ, and is independent ofS, A, and the other parameters of the problem. However, the minimal number of tests does depend on all the parameters of the testing problem, i.e., ε, α,K, and the eigenvalues ofC _μ. In contrast to the worst case setting, it also depends on the dimensiond of the range space ofS andA. This work was done while consulting at Bell Laboratories, and is partially supported by the National Science Foundation and the Air Force Office of Scientific Research.     

Regular centralizers of idempotent transformations

Denote by T(X) the semigroup of full transformations on a set X. For ε∈T(X), the centralizer of ε is a subsemigroup of T(X) defined by C(ε)={α∈T(X):αε=εα}. It is well known that C(id _X )=T(X) is a regular semigroup. By a theorem proved by J.M. Howie in 1966, we know that if X is finite, then the subsemigroup generated by the idempotents of C(id _X ) contains all non-invertible transformations in C(id _X ).     

Direct products of cyclic semigroups an unions of groups

We consider direct productsS×U_eεE G _e=S ₁×…×S _n × U_eεE G _e of non-group finite cyclic semigroupsS _i, 1 ≤i ≤n, and finite unions of finite groups U_eεE G _e We prove that if such a semigroup is isomorphic to another of the same form, sayT×U _fεF H _f =T ₁×…×U _fεF H _f , whereT _j are non-group cyclic semigroups, 1≤j≤l, and U _fεF H _f is a union of groups, thenS is isomorphic toT and U_eεE G _e is isomorphic to U_fεF H _f . We then determine when a finite semigroup has such a decomposition and show how the direct factors can be found.     

S-Classification of Regular Semigroups

On any regular semigroup S, the greatest idempotent pure congruence τ the greatest idempotent separating congruence μ and the least band congruence β are used to give the S-classification of regular semigroups as follows. These congruences generate a sublattice Λ of the congruence lattice C(S) of S. We consider the triples (Λ,K,T), where K and T are the restrictions of the K- and T-relations on C(S) to Λ. Such triples are characterized abstractly and form the objects of a category S whose morphisms are surjective K- and T-preserving homomorphisms subject to a mild condition. The class of regular semigroups is made into a category S whose morphisms are fairly restricted homomorphisms. The main result of the paper is the existence of a representative functor from S to S. The effect of the S-classification on Reilly semigroups and cryptogroups is discussed briefly.     

Pseudo-contractibility and pseudo-amenability of semigroup algebras

In this paper, we characterize pseudo-contractibility of ℓ ¹(S), where S is a uniformly locally finite inverse semigroup. As a consequence, we show that for a Brandt semigroup S=M⁰(G,I),{S={\mathcal{M}}^{0}(G,I),} the semigroup algebra ℓ ¹(S) is pseudo-contractible if and only if G and I are finite. Moreover, we study the notions of pseudo-amenability and pseudo-contractibility of a semigroup algebra ℓ ¹(S) in terms of the amenability of S.     

Instability of Set Recurrence and Green’s Function on Groups with the Liouville Property

Let μ and ν be probability measures on a group Γ and let G _μ and G _ν denote Green’s function with respect to μ and ν. The group Γ is said to admit instability of Green’s function if there are symmetric, finitely supported measures μ and ν and a sequence {x _n } such that G _μ (e, x _n )/G _ν (e,x _n ) →0, and Γ admits instability of recurrence if there is a set S that is recurrent with respect to ν but transient with respect to μ. We give a number of examples of groups that have the Liouville property but have both types of instabilities. Previously known groups with these instabilities did not have the Liouville property.     

Biordered Sets of Regular Semigroups with Inverse Transversals1

In a regular semigroup S, an inverse subsemigroup S° of S is called an inverse transversal of S if S° contains a unique inverse x° of each element x of S. An inverse transversal S° of S is called a Q-inverse transversal of S if S° is a quasi-ideal of S.If S is a regular semigroup with set of idempotents E then E is a biordered set. T.E. Hall obtained a fundamental regular semigroup T_E from the subsemigroup E which is generated by the set of idempotents of a regular semigroup. K.S.S. Nambooripad constructed a fundamental regular semigroup by a regular biordered set abstractly. In this paper, we discuss the properties of the biordered sets of regular semigroups with inverse transversals. This kind of regular biordered sets is called IT-biordered sets. We also describe the fundamental regular semigroup T_E when E is an IT-biordered set. In the sequel, we give the construction of an IT-biordered set by a left regular IT-biordered set and a right regular IT-biordered set.This project has been supported by the Provincial Natural Science Foundation of Guangdong Province, PR China     

Maximizing a congruence with respect to its partition of idempotents

For a congruence σ on a semigroupS a congruence μ(σ) onS, containing σ, is defined such that the semigroupS/σ is fundamental if and only if σ=μ(σ). The congruence μ(σ) is shown to possess maximality properties and for idempotent-surjective semigroups, μ(σ) is the maximum congruence with respect to the partition of the idempotents determined by σ. Thus μ is the maximum idempotent-separating congruence on any idempotent-surjective semigroup. It is shown that μ(μ(σ))=μ(σ). If ρ is another congruence onS, possibly with the same partition of the idempotents as σ, then it is of interest to know when ρ⊆σ (or ρ⊆μ(σ)) implies μ(ρ)⊆μ(σ) or even μ(ρ)=μ(σ). These implications are not true in general but if σ⊆ρ⊆μ(σ) then μ(ρ)⊆μ(σ). IfS is an idempotent-surjective semigroup and ρ and σ have the same partition of the idempotents then μ(ρ)=μ(σ).     

TC semigroups and inflations

An algebraA satisfiesTC (the term condition) if for any and anyn + 1-ary termp.TC algebras have been extensively studied. We previously determined the structure of allTC semigroups. We use this result to show that ifS is aTC semigroup thenS _E = {a ε S | ax is an idempotent for somex ε S} is an inflation ofS _Reg (the set of regular elements ofS) andS _Reg ≅H × A × B whereH is an abelian group,A is a left zero semigroup, andB is a right zero semigroup. As a corollary of this result, we show thatS is a semisimpleTC semigroup iffS ≅H × A × B whereH is an abelian group,A is a left zero semigroup, andB is a right zero semigroup.