Generic automorphisms of the universal partial order

Kuske, Dietrich; Truss, John K.

doi:10.1090/s0002-9939-00-05778-6

Cited by 23 publications

(31 citation statements)

References 10 publications

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“…Our main results are similar to the ones in [10,18] regarding (Q, <) and the universal partial order (although it is not known if the universal partial order has a generic pair); more recently, a new preprint [12] appeared giving similar results on two different structures (the universal ordered boron tree-roughly speaking, a graph theoretic binary tree with a lexicographical order-and the universal ordered poset). The latter's motivation came from a different yet related question of finding an ultrahomogeneous ordered structure whose automorphism group has ample generics and is extremely amenable (in other words, by [8], its age has the Ramsey property).…”

supporting

confidence: 84%

“…We aim to show that the universal countable meet-tree admits a generic automorphism (even though, by Corollary 3.8, we already know it does not admit a generic pair of automorphisms). Very broadly, the proof follows [10]. More precisely, we will find a sufficient condition for a partial automorphism to be an amalgamation base in the class K 1 (where K is the class of finite meet-trees), and in the next section, we will find a cofinal class of automorphisms satisfying this condition, thus showing CAP for K 1 .…”

Section: Now We Need To Show That H( ∧ ) = H( ) ∧ H(mentioning

confidence: 90%

“…Note that the answer to this question in general is negative (since [11] shows that the countable atomless Boolean algebra has ample generics), so it seems that the question should be investigated under the NIP assumption. The obvious candidate, (Q, <), fails: by the works of Hodkinson (unpublished), Truss [18], and the third author (who gave a new proof of this result) [16, Lemma 6.1.1], we know that Aut(Q, <) has no generic pair of automorphisms (although Aut(Q, <) does have a generic automorphism by [10,17]). Meet-trees are partial orders in which for every element a, the set of elements below it is linearly ordered, equipped with a meet function (see Definition 2.11).…”

mentioning

confidence: 99%

“…In Section 3 we prove that the automorphism group of the countable dense tree does not have a generic pair of automorphisms (in fact we give a more general statement, also about trees with bounded arity). In Section 4 we follow [10] and discuss determined finite partial automorphisms in an abstract context, giving a sufficient condition under which they form a class of amalgamation bases. In Section 5 we discuss the possible orbits of partial automorphisms in meet-trees.…”

mentioning

confidence: 99%

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On the Automorphism Group of the Universal Homogeneous Meet-Tree

Kaplan¹,

Rzepecki²,

Siniora³

2021

J. symb. log.

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supporting

confidence: 84%

Section: Now We Need To Show That H( ∧ ) = H( ) ∧ H(mentioning

confidence: 90%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

On the Automorphism Group of the Universal Homogeneous Meet-Tree

Kaplan¹,

Rzepecki²,

Siniora³

2021

J. symb. log.

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“…We next list some examples of structures that admit generic automorphisms. (i) N; (Kruske-Truss [34]) the random poset P ; (Truss [59]) Q, < .…”

Section: Topological Dynamics Ii: Generic Symmetries and Automatic Continuity Phenomenamentioning

confidence: 99%

European Congress of Mathematics Kraków, 2 – 7 July, 2012

2013

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A topological group G is Polish if its topology admits a compatible separable complete metric. Such a group is non-archimedean if it has a basis at the identity that consists of open subgroups. This class of Polish groups includes the profinite groups and (Qp, +) but our main interest here will be on non-locally compact groups.In recent years there has been considerable activity in the study of the dynamics of Polish non-archimedean groups and this has led to interesting interactions between logic, finite combinatorics, group theory, topological dynamics, ergodic theory and representation theory. In this paper I will give a survey of some of the main directions in this area of research.

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The endomorphism monoid of the random poset contains all countable semigroups

Dolinka

2007

Algebra univers.

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The transformation monoid on a countable set (and hence all countable semigroups) embeds in the endomorphism monoid of the countable homogeneous universal poset, the Fraïssé limit of the class of all finite posets.A recurring theme in the theory of transformation semigroups is to find representations of various classes of semigroups by functions having some prescribed properties. For example, there is a significant number of results dealing with endomorphism monoids of first-order structures. In the course of investigating semigroup-theoretical properties of such monoids, it is fairly natural to ask the question of which semigroups admit a representation by endomorphisms of the considered structure.This short note is spurred by conversations I had with my colleague Csaba Szabó [16] at the 2nd Novi Sad Algebraic Conference (NSAC) in July 2005. These conversations followed his short but quite intriguing talk entitled "What does the random poset look like?", and the very title of the talk was reiterated by him and Pawel Idziak at the problem session a few days later. (Clearly enough, there is no unique and clean-cut answer to such a question, as it rather aims at investigating various representations of this interesting object. A recent paper [12] by Hubička and Nešetřil offers a few possible answers in terms of finite graph homomorphisms.)It came as no surprise that R, the random graph (the unique countable homogeneous universal graph) [2,4,5], was discussed during these conversations. Among other things, I mentioned the result by Bonato, Delić and myself [3] that End(R) contains an isomorphic copy of T ℵ0 , the transformation monoid on a countable set. So, this is how we came up with the question of whether the same holds for the random poset [15,7], the unique countable homogeneous universal poset.

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Generic automorphisms of the universal partial order

Cited by 23 publications

References 10 publications

On the Automorphism Group of the Universal Homogeneous Meet-Tree

On the Automorphism Group of the Universal Homogeneous Meet-Tree

European Congress of Mathematics Kraków, 2 – 7 July, 2012

The endomorphism monoid of the random poset contains all countable semigroups

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