On relative ranks of full transformation semigroups

Howie, James; Ruškuc, Nik; Higgins, Peter M.

doi:10.1080/00927879808826160

Cited by 53 publications

(51 citation statements)

References 9 publications

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“…We now consider some of the problems from above and from [8] in a more general setting. Throughout the following, let X be an infinite set and let B X denote the semigroup of all binary relations on X.…”

Section: Binary Relationsmentioning

confidence: 99%

“…We now consider the relative rank of B X modulo some standard subsemigroups, simulating the results in [8]. We begin by finding the relative rank of B X modulo the set of all injective mappings in T X .…”

Section: Corollary 32mentioning

confidence: 99%

“…For β ∈ B X , if im(β) = X then we call the relation surjective. As in [8], where the infinite contraction index is used as a means of distinguishing permutations from maps which are not permutations, we require a property of relations that distinguish maps from relations which are not maps. To this end, for an arbitrary relation α ∈ B X , we define the infinite expansion index to be the cardinality of the set P (α) = {x ∈ dom(α) : |xα| = |X|}, and we denote this cardinal by p(α).…”

Section: Corollary 13 Every Countable Semigroup Is Embeddable In a mentioning

confidence: 99%

“…We also allow ourselves to call this the relative rank of A in S. In [8] the authors showed that if we take S = T X , an infinite full transformation semigroup, then the relative rank of S modulo A is 2 in the cases where A is either the full symmetric group or the set of all idempotents of S. On the other hand, if A is itself a countable set and A ∪ B generates T X , then B must be uncountable as T X is itself uncountable. This prompts the question as to whether it is possible to find a subset A of T X which is of countable relative rank greater than 2.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Countable Versus Uncountable Ranks in Infinite Semigroups of Transformations and Relations

Higgins¹,

Howie²,

Mitchell³

et al. 2003

Proceedings of the Edinburgh Mathematical Society

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The relative rank rank(S : A) of a subset A of a semigroup S is the minimum cardinality of a set B such that A ∪ B = S. It follows from a result of Sierpiński that, if X is infinite, the relative rank of a subset of the full transformation semigroup T X is either uncountable or at most 2. A similar result holds for the semigroup B X of binary relations on X.A subset S of T N is dominated (by U ) if there exists a countable subset U of T N with the property that for each σ in S there exists µ in U such that iσ iµ for all i in N. It is shown that every dominated subset of T N is of uncountable relative rank. As a consequence, the monoid of all contractions in T N (mappings α with the property that |iα − jα| |i − j| for all i and j) is of uncountable relative rank.It is shown (among other results) that rank(B X : T X ) = 1 and that rank(B X : I X ) = 1 (where I X is the symmetric inverse semigroup on X). By contrast, if S X is the symmetric group, rank(B X : S X ) = 2.

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Section: Binary Relationsmentioning

confidence: 99%

Section: Corollary 32mentioning

confidence: 99%

Section: Corollary 13 Every Countable Semigroup Is Embeddable In a mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Countable Versus Uncountable Ranks in Infinite Semigroups of Transformations and Relations

Higgins¹,

Howie²,

Mitchell³

et al. 2003

Proceedings of the Edinburgh Mathematical Society

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show abstract

“…The relative rank of the full transformation semigroup T X , over an infinite set X, modulo various standard subsemigroups was first considered in [7]. It was shown in [11] that the relative rank of T X modulo S X is two. In the same paper it was shown that the relative rank of the set of all idempotent maps on X in T X is, also, two.…”

Section: Introductionmentioning

confidence: 99%

Generating the Full Transformation Semigroup Using Order Preserving Mappings

Higgins¹,

Mitchell

Ruškuc³

2003

Glasgow Math. J.

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Abstract. For a linearly ordered set X we consider the relative rank of the semigroup of all order preserving mappings O X on X modulo the full transformation semigroup T X . In other words, we ask what is the smallest cardinality of a set A of mappings such that O X ∪ A = T X . When X is countably infinite or well-ordered (of arbitrary cardinality) we show that this number is one, while when X = ‫ޒ‬ (the set of real numbers) it is uncountable.2000 Mathematics Subject Classification. 20M20, 06A05.

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Polymorphism clones of homogeneous structures: generating sets, Sierpiński rank, cofinality and the Bergman property

Pech¹,

Pech

2018

Algebra Univers.

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On relative ranks of full transformation semigroups

Cited by 53 publications

References 9 publications

Countable Versus Uncountable Ranks in Infinite Semigroups of Transformations and Relations

Countable Versus Uncountable Ranks in Infinite Semigroups of Transformations and Relations

Generating the Full Transformation Semigroup Using Order Preserving Mappings

Polymorphism clones of homogeneous structures: generating sets, Sierpiński rank, cofinality and the Bergman property

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