Automorphism groups of countable highly homogeneous partially ordered sets

Glass, A. M. W.; McCleary, Stephen H.; Rubin, Matatyahu

doi:10.1007/bf02572390

Cited by 24 publications

(27 citation statements)

References 12 publications

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“…Another possible use of generics would be to streamline the proof of the simplicity of Aut(P, <) given in [1] (where 16 conjugates were in general required to express one non-identity element as a product of conjugates of another). The correct minimum number is probably 3 or 4, and one could establish sufficiency of 4 by showing that for any non-identity g 1 and g 2 , there is h such that g 1 h −1 g 2 h is generic.…”

Section: Further Questionsmentioning

confidence: 99%

“…The two main examples we have in mind are Aut(Q, <) (where we already knew from [11] that there are generics), and the automorphism group of the countable universalhomogeneous partial ordering (P, <) (where we did not). This latter structure has been considered by Schmerl (in the context of his classification of all the countable homogeneous partial orders [10]) and Glass, McCleary and Rubin [1], in studying its automorphism group (principally the verification of its simplicity). The point about these two cases is that we cannot possibly expect to use automorphisms of finite substructures, because all but trivial finite partial automorphisms must have distinct domain and range.…”

Section: Introductionmentioning

confidence: 99%

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Generic automorphisms of the universal partial order

Kuske

Truss

2000

Proc. Amer. Math. Soc.

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Abstract. We show that the countable universal-homogeneous partial order (P, <) has a generic automorphism as defined by the second author, namely that it lies in a comeagre conjugacy class of Aut(P, <). For this purpose, we work with 'determined' partial finite automorphisms that need not be automorphisms of finite substructures (as in the proofs of similar results for other countable homogeneous structures) but are nevertheless sufficient to characterize the isomorphism type of the union of their orbits.

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Section: Further Questionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generic automorphisms of the universal partial order

Kuske

Truss

2000

Proc. Amer. Math. Soc.

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“…We single out, among others, the results of [10,13] and refer to [6,14] for a broader view to the field of automorphisms of homogeneous posets, as well as for a wealthy source of open problems.…”

Section: Dolinkamentioning

confidence: 99%

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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“…Let (X, <) be the countable universal poset and G = Aut(X, <). If g is any nonidentity element of G, then every element of G can be written as a product of at most 16 conjugates of alternately g and its inverse [GMR,Thm. 1'], so G is simple in a very strong sense.…”

Section: 2mentioning

confidence: 99%

“…3.9]. However, G contains strongly embracing elements (that is, elements h that satisfy x < xh for all x ∈ X) [GMR,Cor. 3.3], and products or conjugates of strongly embracing elements are again strongly embracing, so the conjugation-invariant subsemigroup generated by a strongly embracing element is not all of G.…”

Section: 2mentioning

confidence: 99%

Products of similar matrices

Witte¹

1998

Proc. Amer. Math. Soc.

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Abstract. Let A and B be n × n matrices of determinant 1 over a field K, with n > 2 or |K| > 3. We show that if A is not a scalar matrix, then B is a product of matrices similar to A. Analogously, we conjecture that if a and b are elements of a semisimple algebraic group G over a field of characteristic zero, and if there is no normal subgroup of G containing a but not b, then b is a product of conjugates of a. The conjecture is verified for orthogonal groups and symplectic groups, and for all semisimple groups over local fields. Thus, in a connected, semisimple Lie group with finite center, the only conjugationinvariant subsemigroups are the normal subgroups.

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Automorphism groups of countable highly homogeneous partially ordered sets

Cited by 24 publications

References 12 publications

Generic automorphisms of the universal partial order

Generic automorphisms of the universal partial order

The endomorphism monoid of the random poset contains all countable semigroups

Products of similar matrices

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