Two statements about infinite products that are not quite true

Bergman, George M.

doi:10.1090/conm/420/07967

Cited by 8 publications

(10 citation statements)

References 21 publications

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“…Taking complements, we have Σ 1 f −1 ab = Σ 1 , so as Σ 1 is full with respect to U , we can find c ∈ U {Σ1} which agrees on Σ 1 with the inverse of f −1 ab; that is, such that f −1 abc ∈ S (Σ1) . Now (2), applied to the inverse of the latter element, gives us…”

Section: Introduction Notation and Some Lemmas On Full Moietiesmentioning

confidence: 99%

Generating Infinite Symmetric Groups

Bergman

2006

Bull. London Math. Soc.

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137

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Section: Introduction Notation and Some Lemmas On Full Moietiesmentioning

confidence: 99%

Generating Infinite Symmetric Groups

Bergman

2006

Bull. London Math. Soc.

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137

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“…Clearly the first assertion is a case of the second. To prove the latter, let us, for any poset P, write fdown(P ) for the upper semilattice of finite nonempty unions of principal downsets of P. Then one can verify that (2) fdown(P ) ∼ = the upper semilattice freely generated by the poset P.…”

Section: Nonexistence Of Surjectionsmentioning

confidence: 99%

On lattices and their ideal lattices, and posets and their ideal posets

Bergman¹

2008

Tbilisi Math. J.

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For P a poset or lattice, let Id(P ) denote the poset, respectively, lattice, of upward directed downsets in P, including the empty set, and let id(P ) = Id(P ) − {∅}. This note obtains various results to the effect that Id(P ) is always, and id(P ) often, "essentially larger" than P. In the first vein, we find that a poset P admits no <-respecting map (and so in particular, no one-to-one isotone map) from Id(P ) into P, and, going the other way, that an upper semilattice S admits no semilattice homomorphism from any subsemilattice of itself onto Id(S).The slightly smaller object id(P ) is known to be isomorphic to P if and only if P has ascending chain condition. This result is strengthened to say that the only posets P 0 such that for every natural number n there exists a poset Pn with id n (Pn) ∼ = P 0 are those having ascending chain condition. On the other hand, a wide class of cases is noted where id(P ) is embeddable in P.Counterexamples are given to many variants of the statements proved.2000 Mathematics Subject Classification. Primary: 06A06, 06A12, 06B10. Key words and phrases. lattice of ideals of a lattice or semilattice; poset of ideals (upward directed downsets) of a poset; ideals generated by chains.The author is indebted to the referee for many helpful corrections and references. After publication of this note, updates, further references, errata, etc., if found, will be recorded at

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“…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 ∪ {g, h} for some g, h ∈ E. But, by the hypotheses onThis result is proved by a very different method in [3].…”

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confidence: 98%

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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Two statements about infinite products that are not quite true

Cited by 8 publications

References 21 publications

Generating Infinite Symmetric Groups

Generating Infinite Symmetric Groups

On lattices and their ideal lattices, and posets and their ideal posets

Generating Self-Map Monoids of Infinite Sets

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