Semigroup Structure in Stone-čech Compactifications

Pym, J. S.

doi:10.1112/jlms/s2-36.3.421

Cited by 26 publications

(30 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The structure of H ω is that induced by an "oid" as introduced by John Pym [8]. When we say that two structures are "topologically and algebraically isomorphic", we mean that there is one function between them that is both an isomorphism and a homeomorphism.…”

Section: The Semigroups H κmentioning

confidence: 99%

Large rectangular semigroups in Stone-Cech compactifications

Hindman

Strauss

Zelenyuk

2003

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We show that large rectangular semigroups can be found in certain Stone-Čech compactifications. In particular, there are copies of the 2 c × 2 c rectangular semigroup in the smallest ideal of (βN, +), and so, a semigroup consisting of idempotents can be embedded in the smallest ideal of (βN, +) if and only if it is a subsemigroup of the 2 c × 2 c rectangular semigroup. In fact, we show that for any ordinal λ with cardinality at most c, βN contains a semigroup of idempotents whose rectangular components are all copies of the 2 c × 2 c rectangular semigroup and form a decreasing chain indexed by λ + 1, with the minimum component contained in the smallest ideal of βN.As a fortuitous corollary we obtain the fact that there are ≤ L -chains of idempotents of length c in βN. We show also that there are copies of the direct product of the 2 c × 2 c rectangular semigroup with the free group on 2 c generators contained in the smallest ideal of βN.

show abstract

Section: The Semigroups H κmentioning

confidence: 99%

Large rectangular semigroups in Stone-Cech compactifications

Hindman

Strauss

Zelenyuk

2003

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Any commutative standard oid can be considered as⊕ =1 ∞ {1, ∞}\ {(1,1, … ,1)}. We use epithet "standard"to indicate that the index set is ℕ (in [7],oids could have any index set). For , ∈ , < means that < if ∈ and ∈ , and → ∞ for some net ( ) in will mean that for arbitrary ∈ ℕ eventually min > .…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…Thus the product is required to be defined if and only if either or is 1. Obviously, the product in is associative where defined and = ( ) ∪ ( ) whenever is defined in (oids are discussed in [7]). Any commutative standard oid can be considered as⊕ =1 ∞ {1, ∞}\ {(1,1, … ,1)}.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…Civin and Yood [4] shows that the Stone-Čech compactification of a discrete semigroup could be given a semigroup structure, which need not be commutative on and is continuous in the left-hand variable; (that is for fixed ∈ , the map → : → is continuous).Indeed the operation on extends uniquely to , so that contained in it's topological center [5]. Pym [7] introduced the concept of an oid (see Section 2 for precise definition). Oids are important because nearly all semigroups contain them and all oids are oid-isomorphic [6].We shall present our theory in a fairly concrete setting, so that our methods and results will be more readily accessible.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Compact Topological Semigroups Associated With Oids

AMINPOUR¹,

Seilani²

2014

J. Math. Computer Sci.

View full text Add to dashboard Cite

The known theory for a discrete oid shows that how to find a subset ∞ of which is a compact right topological semigroup (see section 2 for details).In this paper we try to find an analogue of almost periodic functions for oids. We discover, new compact semigroups by using a certain subspace of functions ∞ ( ) of ( ) for an oid for which is continuous on ∞ × ( ∪ ∞ ∪ ∞ ),where( ∪ ∞ ∪ ∞ ) is a suitable subspace of for a wide range.Mathematical Society Classification:2010, 54D35.

show abstract

“…Leader, we used a similar expansion to the base −k in [7] to establish that certain natural infinite matrices are not image partition regular. A weak version of digital representation, the notion of oid , was introduced by J. Pym in [13] and is sufficient to derive much of the algebraic structure of the Stone-Čech compactification of N.…”

Section: Introductionmentioning

confidence: 99%

Digital representation of semigroups and groups

2008

View full text Add to dashboard Cite

Abstract. 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, F t = {x t , x 2 t , . . . , x m t −1 t } 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.

show abstract

Semigroup Structure in Stone-čech Compactifications

Cited by 26 publications

References 0 publications

Large rectangular semigroups in Stone-Cech compactifications

Large rectangular semigroups in Stone-Cech compactifications

Compact Topological Semigroups Associated With Oids

Digital representation of semigroups and groups

Contact Info

Product

Resources

About