The extension of the operation in the Stone-Čech compactification of semigroups

For a large class of locally compact semitopological semigroups S, the Stone-Čech compactification β S is a semigroup compactification if and only if S is either discrete or countably compact. Furthermore, for this class of semigroups which are neither discrete nor countably compact, the quotient contains a linear isometric copy of ℓ _∞. These results improve theorems by Baker and Butcher and by Dzinotyiweyi.     

α-Inflatable semigroups

For α∈N with α≥2, we define and characterize α-inflatable semigroups,S and establish that the product (βS/S,·)·(βS/S,·) of Stone-Ĉech remainders does not contain the closure of the minimal ideal of (βS,·), the Stone-Ĉech compactification ofS. From this result, one can easily derive Ruppert's result that the minimal ideal of a compact left-topological semigroup is not necessarily closed. The author gratefully acknowledges support from Delaware State College under Grant No. 6769.     

Oids,finite sums and the structure of the Stone-Čech compactification of a discrete semigroup

The equivalence between oids and semigroups which satisfy a condition involving finite sums is established. Some of the already known results on the structure of Stone-Čech compactifications of discrete semigroups are obtained as immediate consequences. It is also shown that most commutative semigroups contain oids so that oid theory has applications to the Stone-Čech compactifications of many semigroups.     

Connectedness extended to <Emphasis Type="Italic">σ</Emphasis>-frames

We introduce and study the concepts of connectedness and local connectedness in σ-frames. We also consider the local connectedness of the Stone-Čech compactification of a regular σ-frame.      

Right cancellation in βS andUG

Let βS be the Stone-Ĉech compactification of an infinite discrete cancellative semigroupS. The set of points in the growth βS/S at which right cancellation holds in βS is shown to be dense in βS/S. It is then deduced that these type of points also form a dense subset ofUG/G, whenG is a non-compact locally compact abelian group andUG is its uniform compactification.     

Perfect compactifications of frames

Perfect compactifications of frames are introduced. It is shown that the Stone-Čech compactification is an example of such a compactification. We also introduce rim-compact frames and for such frames we define its Freudenthal compactification, another example of a perfect compactification. The remainder of a rim-compact frame in its Freudenthal compactification is shown to be zero-dimensional. It is shown that with the assumption of the Boolean Ultrafilter Theorem the Freudenthal compactification for spaces, as well as the Freudenthal-Morita Theorem for spaces, can be obtained from our frame constructions.     

Free groups in the smallest ideal of β G

Let G be an infinite group with cardinality κ embeddable into a direct sum of countable groups and let β G be the Stone-Čech compactification of G as a discrete semigroup. We show that the structural group of the smallest ideal of β G contains copies of the free group on generators. Supported by NRF grants FA2007041200005 and IFR2008041600015, respectively, and The John Knopfmacher Centre for Applicable Analysis and Number Theory.     

Completely regular paracompact reflection of locales

We give an explicit construction of the completely regular paracompact reflection pL of a completely regular locale L described as a sublocale of the Stone-tech compactification βL of L.     

The closure of the smallest ideal of an ultrafilter semigroup

Let S be a discrete semigroup, let β S be the Stone-Čech compactification of S, and let T be a closed subsemigroup of β S. We characterize ultrafilters from the smallest ideal K(T) of T and from its closure c ℓ  K(T). We show that, for a large class of closed subsemigroups of β S, c ℓ  K(T) is not an ideal of T. This class includes the subsemigroups 0⁺⊂βℝ _d and ℍ _κ ⊂β(⊕ _κ ℤ₂).     

Points very close to the smallest ideal of βS

Given any infinite set of subsets of a discrete semigroupS we determine precisely whether they extend to (neighborhoods of) a point in the smallest two sided ideal of βS, the Stone-Čech compactfication ofS. We utilize this information to study the smallest ideal of (βN, +). The first author gratefully acknowledges support received from the National Science Foundation via grant DMS-9025025     

The smallest ideal of βS is not closed

Let S be an infinite discrete semigroup which can be embedded algebraically into a compact topological group and let βS be the Stone–Čech compactification of S. We show that the smallest ideal of βS is not closed.     

Filters and semigroup properties

The existence of non-fixed, almost translation invariant ultrafilters on any infinite semigroupS satisfying some algebraic properties is established using an ultrafilter approach. The structure of the Stone-?ech compactification of any discrete semigroup is investigated using filters and closed subsets ofßS.     

Strongly Zero-Dimensional Locales

New kinds of strongly zero-dimensional locales are introduced and characterized, which are different from Johnstone's, and almost all the topological properties for strongly zero-dimensional spaces have the pointless localic forms. Particularly, the Stone-Čech compactification of a strongly zero-dimensional locale is strongly zero-dimensional. Received January 21, 1999, Accepted February 1, 2000     

Digital representation of semigroups and groups

A digital representation of a semigroup (S,⋅) is a family 〈F _t 〉 _t∈I , where I is a linearly ordered set, each F _t is a finite non-empty subset of S and every element of S is uniquely representable in the form Π _t∈H   x _t where H is a finite subset of I, each x _t ∈F _t and products are taken in increasing order of indices. (If S has an identity 1, then Π _t∈∅   x _t =1.) A strong digital representation of a group G is a digital representation of G with the additional property that for each t∈I, for some x _t ∈G and some m _t >1 in ℕ where m _t =2 if the order of x _t is infinite, while, if the order of x _t is finite, then m _t is a prime and the order of x _t is a power of m _t . We show that any free semigroup has a digital representation with each | F _t |=1 and that each Abelian group has a strong digital representation. We investigate the problem of whether all groups, or even all finite groups have strong digital representations, obtaining several partial results. Finally, we give applications to the algebra of the Stone-Čech compactification of a discrete group and the weakly almost periodic compactification of a discrete semigroup. Dedicated to Karl Heinrich Hofmann on the occasion of his 75th birthday. Stefano Ferri was partially supported by a research grant of the Faculty of Sciences of Universidad de los Andes. The support is gratefully acknowledged. Neil Hindman acknowledges support received from the National Science Foundation via Grant DMS-0554803.     

Irregular abelian semigroups with weakly amenable semigroup algebra

In this paper we give counterexamples for the open problem, posed by Blackmore (Semigroup Forum 55:359–377, 1987) of whether weak amenability of a semigroup algebra ℓ ¹(S) implies complete regularity of the semigroup S. We present a neat set of conditions on a commutative semigroup (involving concepts well known to those working with semigroups, e.g. the counterexamples are nil and 0-cancellative) which ensure that S is irregular (in fact, has no nontrivial regular subsemigroup), but ℓ ¹(S) is weakly amenable. Examples are then given.     

A note on two equivalence relations on inverse semigroups

Let S be an inverse semigroup. In Rezavand et al. (Semigroup Forum 77:300–305, 2008) and Munn (Proc. Glasgow Math. Assoc. 5:41–48, 1961) two equivalence relations are defined on  S. After describing these relations we show that they coincide.     

N -Fold ?ech Derived Functors and Generalised Hopf Type Formulas

The notion of n-fold Čech derived functors is introduced and studied. This is illustrated using the n-fold Čech derived functors of the nilization functors Z_k. This gives a new purely algebraic method for the investigation of the Brown–Ellis generalised Hopf formula for the higher integral group homology and for its further generalisation. The paper ends with an application to algebraic K-theory. (Received: December 2004)     

Medial permutable semigroups of the first kind

We say that a semigroup S is a permutable semigroup if the congruences of S commute with each other, that is, α○β=β○α is satisfied for all congruences α and β of S. A semigroup is called a medial semigroup if it satisfies the identity axyb=ayxb. The medial permutable semigroups were examined in Proc. Coll. Math. Soc. János Bolyai, vol. 39, pp. 21–39 (1981), where the medial semigroups of the first, the second and the third kind were characterized, respectively. In Atta Accad. Sci. Torino, I-Cl. Sci. Fis. Mat. Nat. 117, 355–368 (1983) a construction was given for medial permutable semigroups of the second [the third] kind. In the present paper we give a construction for medial permutable semigroups of the first kind. We prove that they can be obtained from non-archimedean commutative permutable semigroups (which were characterized in Semigroup Forum 10, 55–66, 1975). Research supported by the Hungarian NFSR grant No T042481 and No T043034.     

Ring semigroups whose subsemigroups form a chain

A multiplicative semigroup S is called a ring semigroup if an addition may be defined on S so that (S,+,⋅) is a ring. Such semigroups have been well-studied in the literature (see Bell in Words, Languages and Combinatorics, pp. 24–31, World Scientific, Singapore, 1994; Jones in Semigroup Forum 47(1):1–6, 1993; Jones and Ligh in Semigroup Forum 17(2):163–173, 1979). In this note, we use Mihăilescu’s Theorem (formerly Catalan’s Conjecture) to characterize the ring semigroups whose subsemigroups containing 0 form a chain with respect to set inclusion.     

Certain constructions of asymptotic analysis related to the Stone-Čcech compactification

The paper considers the representations of attraction sets in topological spaces and their relations with the relaxation of accessibility problems under the conditions of sequentially relaxed constraints. The author studies the structure of approximate (in essence, asymptotic) solutions and generalized elements and establishes the possibility of their real identification for a certain version (related to the Stone-Čech compactification and the Wallman relaxation) of relaxation of the initial problem. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 26, Nonlinear Dynamics, 2005.