Remarks on weak amalgamation and large conjugacy classes in non-archimedean groups

Weak Amalgamation ◽

Countable Structure

AbstractWe study the notion of weak amalgamation in the context of diagonal conjugacy classes. Generalizing results of Kechris and Rosendal, we prove that for every countable structure M, Polish group G of permutations of M, and $$n \ge 1$$ n ≥ 1 , G has a comeager n-diagonal conjugacy class iff the family of all n-tuples of G-extendable bijections between finitely generated substructures of M, has the joint embedding property and the weak amalgamation property. We characterize limits of weak Fraïssé classes that are not homogenizable. Finally, we investigate 1- and 2-diagonal conjugacy classes in groups of ball-preserving bijections of certain ordered ultrametric spaces.

Generic expansions of ω-categorical structures and semantics of generalized quantifiers

10.2307/2586500 ◽

1999 ◽

Vol 64 (2) ◽

pp. 775-789 ◽

Cited By ~ 5

Author(s):

A. A. Ivanov

Keyword(s):

Complete Metric Space ◽

Amalgamation Property ◽

General Context ◽

Generalized Quantifiers ◽

Joint Embedding ◽

Categorical Structure ◽

Countably Infinite ◽

Countable Structure

Let M be a countably infinite ω-categorical structure. Consider Aut(M) as a complete metric space by defining d(g, h) = Ω{2−n: g (xn) ≠ h(xn) or g−1 (xn) ≠ h−1 (xn)} where {xn : n ∈ ω} is an enumeration of M An automorphism α ∈ Aut(M) is generic if its conjugacy class is comeagre. J. Truss has shown in [11] that if the set P of all finite partial isomorphisms contains a co-final subset P1 closed under conjugacy and having the amalgamation property and the joint embedding property then there is a generic automorphism. In the present paper we give a weaker condition of this kind which is equivalent to the existence of generic automorphisms. Really we give more: a characterization of the existence of generic expansions (defined in an appropriate way) of an ω-categorical structure. We also show that Truss' condition guarantees the existence of a countable structure consisting of automorphisms of M which can be considered as an atomic model of some theory naturally associated to M. We do it in a general context of weak models for second-order quantifiers.The author thanks Ludomir Newelski for pointing out a mistake in the first version of Theorem 1.2 and for interesting discussions. Also, the author is grateful to the referee for very helpful remarks.

AMALGAMABLE DIAGRAM SHAPES

10.1017/jsl.2018.87 ◽

2019 ◽

Vol 84 (1) ◽

pp. 88-101

Author(s):

RUIYUAN CHEN

Keyword(s):

Closed Subset ◽

Amalgamation Property ◽

Finitely Generated ◽

Finite Poset ◽

Simple Rules ◽

Joint Embedding ◽

Image Position ◽

Simply Connected

AbstractA category has the amalgamation property (AP) if every pushout diagram has a cocone, and the joint embedding property (JEP) if every finite coproduct diagram has a cocone. We show that for a finitely generated category I, the following are equivalent: (i) every I-shaped diagram in a category with the AP and the JEP has a cocone; (ii) every I-shaped diagram in the category of sets and injections has a cocone; (iii) a certain canonically defined category ${\cal L}\left( {\bf{I}} \right)$ of “paths” in I has only idempotent endomorphisms. When I is a finite poset, these are further equivalent to: (iv) every upward-closed subset of I is simply-connected; (v) I can be built inductively via some simple rules. Our proof also shows that these conditions are decidable for finite I.

A Note on Saturated Models for Many-Valued Logics

Mathematica Slovaca ◽

10.1515/ms-2015-0053 ◽

2015 ◽

Vol 65 (4) ◽

Author(s):

Tommaso Flaminio ◽

Matteo Bianchi

Keyword(s):

Representation Theorem ◽

Amalgamation Property ◽

Short Paper ◽

Fuzzy Logics ◽

Saturated Models ◽

Joint Embedding ◽

Joint Embedding Property

AbstractIn this short paper we will discuss on saturated and κ-saturated models of many-valued (t-norm based fuzzy) logics. Using these peculiar structures we show a representation theorem à la Di Nola for several classes of algebras including MV, Gödel, product, BL, NM and WNM-algebras. Then, still using (κ)-saturated algebras, we finally show that some relevant subclasses of algebras related to many-valued logics also enjoy the joint embedding property and the amalgamation property.

Model companions for finitely generated universal Horn classes

10.2307/2274092 ◽

1984 ◽

Vol 49 (1) ◽

pp. 68-74 ◽

Cited By ~ 8

Author(s):

Stanley Burris

Keyword(s):

Finitely Generated ◽

Model Companion ◽

Universal Horn Class ◽

Finite Structure ◽

Finite Structures ◽

Structure Theorems ◽

Joint Embedding ◽

Primitive Recursive

AbstractIn an earlier paper we proved that a universal Horn class generated by finitely many finite structures has a model companion. If the language has only finitely many fundamental operations then the theory of the model companion admits a primitive recursive elimination of quantifiers and is primitive recursive. The theory of the model companion is ℵ0-categorical iff it is complete iff the universal Horn class has the joint embedding property iff the universal Horn class is generated by a single finite structure. In the last section we look at structure theorems for the model companions of universal Horn classes generated by functionally complete algebras, in particular for the cases of rings and groups.

ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972721000368 ◽

2021 ◽

pp. 1-5

Author(s):

SH. RAHIMI ◽

Z. AKHLAGHI

Keyword(s):

Finite Group ◽

Normal Subgroup ◽

Conjugacy Class ◽

Regular Graph ◽

Simple Graph ◽

Conjugacy Classes ◽

A Finite Group

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

Finitely generic models of TUH, for certain model companionable theories T

10.2307/2274316 ◽

1985 ◽

Vol 50 (3) ◽

pp. 604-610

Author(s):

Francoise Point

Keyword(s):

Discriminator Variety ◽

Generic Models ◽

Elementary Class ◽

Ordered Groups ◽

Existentially Closed ◽

Closed Element ◽

Starting Point ◽

Joint Embedding ◽

Generic Elements

The starting point of this work was Saracino and Wood's description of the finitely generic abelian ordered groups [S-W].We generalize the result of Saracino and Wood to a class ∑UH of subdirect products of substructures of elements of a class ∑, which has some relationships with the discriminator variety V(∑t) generated by ∑. More precisely, let ∑ be an elementary class of L-algebras with theory T. Burris and Werner have shown that if ∑ has a model companion then the existentially closed models in the discriminator variety V(∑t) form an elementary class which they have axiomatized. In general it is not the case that the existentially closed elements of ∑UH form an elementary class. For instance, take for ∑ the class ∑0 of linearly ordered abelian groups (see [G-P]).We determine the finitely generic elements of ∑UH via the three following conditions on T:(1) There is an open L-formula which says in any element of ∑UH that the complement of equalizers do not overlap.(2) There is an existentially closed element of ∑UH which is an L-reduct of an element of V(∑t) and whose L-extensions respect the relationships between the complements of the equalizers.(3) For any models A, B of T, there exists a model C of TUH such that A and B embed in C.(Condition (3) is weaker then “T has the joint embedding property”. It is satisfied for example if every model of T has a one-element substructure. Condition (3) implies that ∑UH has the joint embedding property and therefore that the class of finitely generic elements of ∑UH is complete.)

THE CONJUGACY CLASSES OF SOME FINITE METABELIAN GROUPS AND THEIR RELATED GRAPHS

Jurnal Teknologi ◽

10.11113/jt.v79.7608 ◽

2016 ◽

Vol 79 (1) ◽

Author(s):

Nor Haniza Sarmin ◽

Ain Asyikin Ibrahim ◽

Alia Husna Mohd Noor ◽

Sanaa Mohamed Saleh Omer

Keyword(s):

Graph Theory ◽

Conjugacy Class ◽

Dihedral Group ◽

Clique Number ◽

Conjugacy Classes ◽

Quaternion Group ◽

Metabelian Groups ◽

Dominating Number ◽

Graph Properties ◽

Class Graph

In this paper, the conjugacy classes of three metabelian groups, namely the Quasi-dihedral group, Dihedral group and Quaternion group of order 16 are computed. The obtained results are then applied to graph theory, more precisely to conjugate graph and conjugacy class graph. Some graph properties such as chromatic number, clique number, dominating number and independent number are found.

Prefrattini Subgroups and Cover-Avoidance Properties in 𝔘-Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1975-091-1 ◽

1975 ◽

Vol 27 (4) ◽

pp. 837-851 ◽

Cited By ~ 4

Author(s):

M. J. Tomkinson

Keyword(s):

Conjugacy Class ◽

Soluble Group ◽

Saturated Formation ◽

Conjugacy Classes ◽

Pronormal Subgroup ◽

Finite Soluble Group ◽

Factors Associated ◽

Chief Factors

W. Gaschutz [5] introduced a conjugacy class of subgroups of a finite soluble group called the prefrattini subgroups. These subgroups have the property that they avoid the complemented chief factors of G and cover the rest. Subsequently, these results were generalized by Hawkes [12], Makan [14; 15] and Chambers [2]. Hawkes [12] and Makan [14] obtained conjugacy classes of subgroups which avoid certain complemented chief factors associated with a saturated formation or a Fischer class. Makan [15] and Chambers [2] showed that if W, D and V are the prefrattini subgroup, 𝔍-normalizer and a strongly pronormal subgroup associated with a Sylow basis S, then any two of W, D and V permute and the products and intersections of these subgroups have an explicit cover-avoidance property.

Numbers Of Conjugacy Class Sizes And Derived Lengths for A-Groups

Canadian Mathematical Bulletin ◽

10.4153/cmb-1996-041-6 ◽

1996 ◽

Vol 39 (3) ◽

pp. 346-351 ◽

Cited By ~ 1

Author(s):

Mary K. Marshall

Keyword(s):

Conjugacy Class ◽

Solvable Group ◽

Conjugacy Classes ◽

Finite Solvable Group ◽

Conjugacy Class Sizes ◽

Sylow Subgroups ◽

Derived Length ◽

Completely Reducible

AbstractAn A-group is a finite solvable group all of whose Sylow subgroups are abelian. In this paper, we are interested in bounding the derived length of an A-group G as a function of the number of distinct sizes of the conjugacy classes of G. Although we do not find a specific bound of this type, we do prove that such a bound exists. We also prove that if G is an A-group with a faithful and completely reducible G-module V, then the derived length of G is bounded by a function of the number of distinct orbit sizes under the action of G on V.

Torsion elements of order 𝑝2 in the Nottingham group

Journal of Group Theory ◽

10.1515/jgth-2019-0114 ◽

2020 ◽

Vol 23 (3) ◽

pp. 489-502

Author(s):

Chun Yin Hui ◽

Krishna Kishore

Keyword(s):

Finite Field ◽

Conjugacy Class ◽

Upper Bound ◽

Conjugacy Classes ◽

Torsion Element ◽

Exact Number

AbstractLet κ be a characteristic p finite field of q elements and {\mathfrak{N}_{\kappa}} the Nottingham group over κ. Lubin associated to every conjugacy class of torsion element of {\mathfrak{N}_{\kappa}} a type. We establish an upper bound {B(q;l,m)} on the number of conjugacy classes of order {p^{2}} torsion elements u of {\mathfrak{N}_{\kappa}} of type {\langle l,m\rangle}. In the case where {l<p}, the bound {B(q;l,m)} is the exact number of conjugacy classes. Moreover, we give a criterion on when u and {u^{n}} are conjugate.