Chain decompositions of ordered semigroups

Characterizations of ordered semigroups which can be decomposed into (natural ordered) chains of ω -simple ordered semigroups are given, where ω -simple ordered semigroups are ξ _l (ξ _t ) -simple, left (t -) simple, L _n (H _n ) -simple, l (t )-archimedean and nil-extensions of left (t -) simple ordered semigroups, respectively. As a generalization of the theory of Clifford semigroups (without orders) to ordered semigroups, ordered semigroups which are semilattices of t -simple subsemigroups are characterized.     

Laplace transforms of vector-valued functions with growth <Emphasis Type="Italic">ω</Emphasis> and semigroups of operators

For an arbitrary continuous, increasing function ω : [0, ∈fty) → C of finite exponential type, we characterize Laplace-Stieltjes transforms of Banach-space-valued functions that are O(ω) , and use this to establish a Hille-Yosida type theorem for strongly continuous semigroups that are O(ω) . Corollaries include characterizing generators of strongly continuous semigroups that are , for d > 1 , in addition to the already known examples of exponential or polynomial growth. January 27, 1999     

Trees and the bireflection property

The group of automorphisms of a tree (partially ordered set where the set of predecessors of an element is well ordered) with no infinite levels enjoys the property that every member is a product of two elements of order ≦2. It is shown that this property—called the bireflection property—fails for some trees having infinite levels. In fact, every subtree of a treeT has the the bireflection property if and only if the tree of all zero-one sequences of length ≦ω with finitely many ones is not embeddable inT.     

Chains of baire class 1 functions and various notions of special trees

Following Laczkovich we consider the partially ordered setB ₁(ℝ) of Baire class 1 functions endowed with the pointwise order, and investigate the order types of the linearly ordered subsets. Answering a question of Komjáth and Kunen we show (inZFC) that special Aronszajn lines are embeddable intoB ₁(ℝ). We also show that under Martin's Axiom a linearly ordered set ℒ with |ℒ| < 2^ω is embeddable intoB ₁(ℝ) iff ℒ does not contain a copy of ω₁ or ω _* ¹ . We present aZFC example of a linear order of size 2^ω showing that this characterisation is not valid for orders of size continuum. These results are obtained using the notion of a compact-special tree; that is, a tree that is embeddable into the class of compact subsets of the reals partially ordered under reverse inclusion. We investigate how this notion is related to the well-known notion of an ℝ-special tree and also to some other notions of specialness. Partially supported by Hungarian Scientific Foundation grant no. 37758, 49786 and F 43620. The second author's research for this paper was partially supported by NSERC of Canada.     

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.     

On quasi-prime,weakly quasi-prime left ideals in ordered-Γ-semigroups

In this paper we obtain and establish some important results in ordered Γ-semigroups extending and generalizing those for semigroups given in [PETRICH, M.: Introduction to Semigroups, Merill, Columbus, 1973] and for ordered semigroups from [KEHAYOPULU, N.: On weakly prime ideals of ordered semigroups, Math. Japon. 35 (1990), 1051–1056], [KEHAYOPULU, N.: On prime, weakly prime ideals in ordered semigroups, Semigroup Forum 44 (1992), 341–346] and [XIE, X. Y.—WU, M. F.: On quasi-prime, weakly quasi-prime left ideals in ordered semigroups, PU.M.A. 6 (1995), 105–120]. We introduce and give some characterizations about the quasi-prime and weakly quasi-prime left ideals of ordered-Γ-semigroups. We also introduce the concept of weakly m-systems in ordered Γ-semigroups and give some characterizations of the quasi-prime and weakly quasi-prime left ideals by weakly m-systems.     

Strongly meager sets can be quite big

AssumeCH. There exists a strongly meager setX⊆2^ω and a continuous functionF: 2^ω → 2^ω such thatF″ (X)=2^ω. The analogous statement for the strong measure zero, the notion dual to strongly meager, is false. The first author was partially supported by NSF grant DMS 9971282 and the Alexander von Humboldt Foundation. The second author was partially supported by grant BW 5100-5-0231-2.     

On strongly-non-reflexive groups

A strong negative answer is given to the old question of whether every dual group is reflexive. Using ◊_ω1 a groupA is constructed so thatA, A*, A**, andA*** are weakly ω₁-separable groups of cardinalityω ₁ andA* is not isomorphic toA***. Research partially supported by NSF Grant No. DMS-8400451. Research partially supported by NSERC Grant No. A8948.     

Ideal extensions of lattices

Following the well-known Schreier extension of groups, the (ideal) extension of semigroups (without order) have been first considered by A. H. Clifford in Trans. Amer. Math. Soc. 68 (1950), with a detailed exposition of the theory in the monographs of Clifford-Preston and Petrich. The main theorem of the ideal extensions of ordered semigroups has been considered by Kehayopulu and Tsingelis in Comm. Algebra 31 (2003). It is natural to examine the same problem for lattices. Following the ideal extensions of ordered semigroups, in this paper we give the main theorem of the ideal extensions of lattices. Exactly as in the case of semigroups (ordered semigroups), we approach the problem using translations. We start with a lattice L and a lattice K having a least element, and construct (all) the lattices V which have an ideal L′ which is isomorphic to L and the Rees quotient V|L′ is isomorphic to K. Conversely, we prove that each lattice which is an extension of L by K can be so constructed. An illustrative example is given at the end. The text was submitted by the author in English.     

When Will Every Maximal <Emphasis Type="BoldItalic">F</Emphasis>-free Subposet Contain a Maximal Element?

Let F be a partially ordered set (poset). A poset P is called F-free if P contains no subposet isomorphic to F. A finite poset F is said to have the maximal element property if every maximal F-free subposet of any finite poset P contains a maximal element of P. It is shown that a poset F with at least two elements has the maximal element property if and only if F is an antichain or F ≅ 2 + 2.     

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) _t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space C_b(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on C_b(Ω). We also prove that the class of bi-continuous semigroups on C_b(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.     

Free discontinuity problems with unbounded data: The two dimensional case

We prove the existence of a minimizing pair for the functionalG defined for every closed setK ⊂R ² and for every functionu ∈C ¹(ω/K) by where ω is an open set inR ², λ, μ>0,q≥1,g ∈L ^q (ω) ∩L ^p (ω) withp>2q andH ¹ is the 1-dimensional Hausdorff measure.     

Splitting squares

We show that for every Borel function f:[2^ω] ⁿ → 2^ω there exists a closed countably splitting family A such that f ↾ [A] ⁿ omits a perfect set of values in 2 ^ω . Work partially supported by DFG grant SP 683.     

Markov—Bernstein-Type Inequality for Trigonometric Polynomials with Respect to Doubling Weights on [-ω,ω]

   Abstract. Various important, weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most cases this minimal assumption is the doubling condition. Here, based on a recently proved Bernstein-type inequality by D. S. Lubinsky, we establish Markov—Bernstein-type inequalities for trigonometric polynomials with respect to doubling weights on [-ω,ω] . Namely, we show the theorem below. Theorem Let p ∈ [1,∞) and ω ∈ (0, 1/2] . Suppose W is a weight function on [-ω,ω] such that W(ω cos t) is a doubling weight. Then there is a constant C depending only on p and the doubling constant L so that

<Emphasis Type="Italic">Eω</Emphasis>-Clifford congruences and <Emphasis Type="Italic">Eω</Emphasis>-<Emphasis Type="Italic">E</Emphasis>-reflexive congruences on an inverse semigroup

This paper enlarges the list of properties of the congruence sequences starting from the universal relation and successively performing the operations of lower T and lower K. Two classes of inverse semigroups, namely Eω-Clifford semigroups and Eω-E-reflexive semigroups, are studied. ((ω _T ) _K ) _T and (((ω _T ) _K ) _T ) _K are found to be the least Eω-Clifford and Eω-E-reflexive congruences on S, respectively. Characterizations of the congruences ((ω _T ) _K ) _T and (((ω _T ) _K ) _T ) _K are developed. The lattices of all Eω-Clifford and Eω-E-reflexive congruences are also studied.     

Countably Compact Paratopological Groups

By means of the theory of bispaces we show that a countably compact T₀ paratopological group (G, τ) is a topological group if and only if (G, τ ∨ τ^-1) is ω-bounded (here τ^-1 is the conjugate topology of τ). Our approach is premised on the fact that every paratopological countably compact paratopological group is a Baire space and on the notion of a 2-pseudocompact space. We also prove that every ω-bounded (respectively, topologically periodic) Baire paratopological group admits a weaker Hausdorff group topology. In particular, ω-bounded (respectively, topologically periodic) 2-pseudocompact (so, also countably compact) paratopological groups enjoy this property. Some topological properties turning countably compact topological semigroups into topological groups are presented and some open questions are posed.     

An <Emphasis Type="Italic">ω</Emphasis>-category with all Duals is an <Emphasis Type="Italic">ω</Emphasis>-groupoid

We make a definition of ω-precategory which should underlie any definition of weak ω-category. We make a precise definition of pseudo-invertible cells in this setting. We show that in an ω-precategory with all weak duals, every cell is pseudo-invertible. We deduce that in any “sensible” theory of ω-categories, an ω-category with all weak duals is an ω-groupoid. We discuss various examples and open questions involving higher-dimensional tangles and cobordisms.      

Generation of analytic semigroups in theL p topology by elliptic operators inR n

Strongly elliptic differential operators with (possibly) unbounded lower order coefficients are shown to generate analytic semigroups of linear operators onL ^p(R ⁿ ), 1≦p≦∞. An explicit characterization of the domain is given for 1<p<∞. An application to parabolic problems is also included. This work has been partially supported by the Research Funds of the Ministero della Pubblica Istruzione. The authors are members of GNAFA (Consiglio Nazionale delle Ricerche).     

Membership of A ∨ G for classes of finite weakly abundant semigroups

We consider the question of membership of A ∨ G, where A and G are the pseudovarieties of finite aperiodic semigroups, and finite groups, respectively. We find a straightforward criterion for a semigroup S lying in a class of finite semigroups that are weakly abundant, to be in A ∨ G. The class of weakly abundant semigroups contains the class of regular semigroups, but is much more extensive; we remark that any finite monoid with semilattice of idempotents is weakly abundant. To study such semigroups we develop a number of techniques that may be of interest in their own right.