Generating continuous mappings with Lipschitz mappings

Cichoń, Jacek; Mitchell, James D.; Morayne, Michał

doi:10.1090/s0002-9947-06-04026-8

Cited by 11 publications

(6 citation statements)

References 12 publications

(9 reference statements)

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“…If T is a subsemigroup of a semigroup S, then we denote by rank(S : T ) the least cardinality of a subset U of S such that T ∪ U generates S. Clearly, if S has Sierpiński rank m ∈ N and T is any subsemigroup of S, then rank(S : T ) m or rank(S : T ) > ℵ 0 . The cardinal rank(S : T ) is referred to as the relative rank of T in S; see [7,20,30]. The following lemma gives an upper bound for the Sierpiński rank of a semigroup in terms of the relative rank and Sierpiński rank of its subsemigroups.…”

Section: Preliminaries and Generalitiesmentioning

confidence: 99%

Generating countable sets of surjective functions

Mitchell

Péresse

2011

Fund. Math.

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We prove that any countable set of surjective functions on an infinite set of cardinality ℵn with n ∈ N can be generated by at most n 2 /2 + 9n/2 + 7 surjective functions of the same set; and there exist n 2 /2 + 9n/2 + 7 surjective functions that cannot be generated by any smaller number of surjections. We also present several analogous results for other classical infinite transformation semigroups such as the injective functions, the Baer-Levi semigroups, and the Schützenberger monoids.

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Section: Preliminaries and Generalitiesmentioning

confidence: 99%

Generating countable sets of surjective functions

Mitchell

Péresse

2011

Fund. Math.

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“…As S(Y ) wr S(Z) is a proper submonoid of each of the monoids T(Y ) wr T(Z), T(Y ) wr S(Z) and S(Y ) wr T(Z), equation ( 8) must be valid. In order to show (7), it suffices to prove that rank (T(Y ) wr T(Z) : S(Y ) wr S(Z)) > 1. Suppose that γ ∈ T(Y ) wr T(Z) such that S(Y ) wr S(Z) ∪ {γ} = T(Y ) wr T(Z).…”

Section: Relative Rank Of Semigroupsmentioning

confidence: 99%

“…Finally, in recent years, the notion of relative rank has been subjected to extensive research (see for example [1,7,13,11,12,16]). Relative rank is a useful notion when dealing with finite semigroups (see Lemma 3.1), and it is crucial when dealing with uncountable semigroups.…”

Section: Introductionmentioning

confidence: 99%

The rank of the endomorphism monoid of a uniform partition

Araújo

Schneider

2008

Semigroup Forum

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The rank of a semigroup is the cardinality of a smallest generating set. In this paper we compute the rank of the endomorphism monoid of a non-trivial uniform partition of a finite set, that is, the semigroup of those transformations of a finite set that leave a non-trivial uniform partition invariant. That involves proving that the rank of a wreath product of two symmetric groups is two and then use the fact that the endomorphism monoid of a partition is isomorphic to a wreath product of two full transformation semigroups. The calculation of the rank of these semigroups solves an open question.

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“…These authors and Mitchell also exhibit various submonoids that are ≈ E in [7]. Related questions, but with other kinds of objects in place of E, are discussed in [5], [7], and [11], as well as in papers referenced therein.…”

Section: Introductionmentioning

confidence: 95%

Generating Self-Map Monoids of Infinite Sets

Mesyan

2007

Semigroup Forum

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Let Ω be a countably infinite set, S = Sym(Ω) the group of permutations of Ω, and E = Self(Ω) the monoid of self-maps of Ω. Given two subgroups G 1 , G 2 ⊆ S, let us write G 1 ≈ S G 2 if there exists a finite subset U ⊆ S such that the groups generated by G 1 ∪ U and G 2 ∪ U are equal. Bergman and Shelah showed that the subgroups which are closed in the function topology on S fall into exactly four equivalence classes with respect to ≈ S . Letting ≈ denote the obvious analog of ≈ S for submonoids of E, we prove an analogous result for a certain class of submonoids of E, from which the theorem for groups can be recovered. Along the way, we show that given two subgroups G 1 , G 2 ⊆ S which are closed in the function topology on S, we have GLet Ω be an arbitrary infinite set, and set E = Self(Ω), the monoid of self-maps of Ω. Elements of E will be written to the right of their arguments. If U ⊆ E is a subset, then we will write U to denote the submonoid generated by U. The cardinality of a set Γ will be denoted by |Γ|. If Σ ⊆ Ω and U ⊆ E are subsets, letDefinition 1. Let M be a monoid that is not finitely generated. Then the cofinality c(M) of M is the least cardinal κ such that M can be expressed as the union of an increasing chain of κ proper submonoids.The main goal of this section is to show that c(E) > |Ω|, which will be needed later on.This section is modeled on Sections 1 and 2 of [2], where analogous statements are proved for the group of all permutations of Ω. Ring-theoretic analogs of these statements are proved in [10].Lemma 2. Let U ⊆ E and Σ ⊆ Ω be such that |Σ| = |Ω| and the set of self-maps of Σ induced by U {Σ} is all of Self(Σ). Then E = gUh, for some g, h ∈ E.Proof. Let g ∈ E be a map that takes Ω bijectively to Σ, and let h ∈ E be a map whose restriction to Σ is the right inverse of g. Then E = gU {Σ} h.We will say that Σ ⊆ Ω is a moiety if |Σ| = |Ω| = |Ω\Σ|. A moiety Σ ⊆ Ω is called full with respect to U ⊆ E if the set of self-maps of Σ induced by members of U {Σ} is all of Self(Σ). The following two results are modeled on group-theoretic results in [9].Lemma 3 (cf. [2, Lemma 3]). Let (U i ) i∈I be any family of subsets of E such that i∈I U i = E and |I| ≤ |Ω|. Then Ω contains a full moiety with respect to some U i .Proof. Since |Ω| is infinite and |I| ≤ |Ω|, we can write Ω as a union of disjoint moieties Σ i , i ∈ I. Suppose that there are no full moieties with respect to U i for any i ∈ I. Then in particular, Σ i is not full with respect to U i for any i ∈ I. Hence, for every i ∈ I there exists a map f i ∈ Self(Σ i ) which is not the restriction to Σ i of any member of (U i ) {Σ i } . Now, if we take f ∈ E to be the map whose restriction to each Σ i is f i , then f is not in U i for any i ∈ I, contradicting i∈I U i = E. Proposition 4. c(E) > |Ω|.Proof. Suppose that (M i ) i∈I is a chain of submonoids of E such that i∈I M i = E and |I| ≤ |Ω|. We will show that E = M i for some i ∈ I.By the preceding lemma, Ω contains a full moiety with respect to some M i . Thus, Lemma 2 implies that E = M i ∪ ...

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Generating continuous mappings with Lipschitz mappings

Cited by 11 publications

References 12 publications

Generating countable sets of surjective functions

Generating countable sets of surjective functions

The rank of the endomorphism monoid of a uniform partition

Generating Self-Map Monoids of Infinite Sets

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