scholarly journals INDEPENDENCE IN GENERIC INCIDENCE STRUCTURES

2019 ◽  
Vol 84 (02) ◽  
pp. 750-780
Author(s):  
GABRIEL CONANT ◽  
ALEX KRUCKMAN
Keyword(s):  
Algebraic Independence ◽  
Incidence Structure ◽  
Projective Planes ◽  
Saturated Model ◽  
Prime Model ◽  
Incidence Structures ◽  
Finite Projective Planes ◽  
Existentially Closed ◽  
Image Position ◽  
Open Question

AbstractWe study the theory Tm,n of existentially closed incidence structures omitting the complete incidence structure Km,n, which can also be viewed as existentially closed Km,n-free bipartite graphs. In the case m = n = 2, this is the theory of existentially closed projective planes. We give an $\forall \exists$-axiomatization of Tm,n, show that Tm,n does not have a countable saturated model when m, n ≥ 2, and show that the existence of a prime model for T2,2 is equivalent to a longstanding open question about finite projective planes. Finally, we analyze model theoretic notions of complexity for Tm,n. We show that Tm,n is NSOP1, but not simple when m, n ≥ 2, and we show that Tm,n has weak elimination of imaginaries but not full elimination of imaginaries. These results rely on combinatorial characterizations of various notions of independence, including algebraic independence, Kim independence, and forking independence.

Incompleteness of (nq + n - q - 2, n)-arcs in finite projective planes of even order

1982 ◽  
Vol 91 (1) ◽  
pp. 1-8
Author(s):  
B. J. Wilson
Keyword(s):  
Projective Planes ◽  
Finite Projective Planes ◽  
Even Order ◽  
Image Position

1. It was shown by Barlotti (1) that the number k of points on a (k, n)-arc of a finite projective piane of order qand that if q ≢ 0 (mod n) then for n ≥ 3.

A Skew Hadamard Matrix of Order 52

1969 ◽  
Vol 21 ◽  
pp. 1319-1322 ◽  
Author(s):  
D. Blatt ◽  
G. Szekeres
Keyword(s):  
Hadamard Matrix ◽  
Projective Planes ◽  
Identity Matrix ◽  
Finite Projective Planes ◽  
Image Position ◽  
Square Matrix

1. A Hadamard (H-) matrix H = (hij) of order n is an n × n square matrix satisfying the conditionsfor all i, j ≦ n. A skew H-matrix is an H-matrix of the formwhere I is the identity matrix and S’ the transpose of 5. In particular,Skew H-matrices have applications in the theory of finite projective planes (2) and tournaments (4), also in the construction of H-matrices of certain orders. For example, if there is a skew H-matrix of order n, then there is an H-matrix of order n(n – 1) (Williamson, see (1, p. 213)).

Existentially closed central extensions of locally finite p-groups

1986 ◽  
Vol 100 (2) ◽  
pp. 281-301 ◽  
Author(s):  
Felix Leinen ◽  
Richard E. Phillips
Keyword(s):  
Central Extensions ◽  
Locally Finite ◽  
Existentially Closed ◽  
Image Position

Throughout, p will be a fixed prime, and will denote the class of all locally finite p-groups. For a fixed Abelian p-group A, we letwhere ζ(P) denotes the centre of P. Notice that A is not a class in the usual group-theoretic sense, since it is not closed under isomorphisms.

Automorphism groups of arithmetically saturated models

2006 ◽  
Vol 71 (1) ◽  
pp. 203-216 ◽  
Author(s):  
Ermek S. Nurkhaidarov
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Open Subgroup ◽  
Peano Arithmetic ◽  
Standard System ◽  
Permutation Groups ◽  
Saturated Model ◽  
Saturated Models ◽  
Homogeneous Set ◽  
Image Position

In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that if M is a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2 be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then SSy(M1) = SSy(M2).We show that if M is a countable arithmetically saturated of Peano Arithmetic, then Aut(M) can recognize if some maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. That fact is used to show the main theorem:Let M1, M2be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then for every n < ωHere RT2n is Infinite Ramsey's Theorem stating that every 2-coloring of [ω]n has an infinite homogeneous set. Theorem 0.2 shows that for models of a false arithmetic the converse of Kossak-Schmerl Theorem 0.1 is not true. Using the results of Reverse Mathematics we obtain the following corollary:There exist four countable arithmetically saturated models of Peano Arithmetic such that they have the same standard system but their automorphism groups are pairwise non-isomorphic.

scholarly journals Embedding (r, 1)-designs in finite projective planes

1977 ◽  
Vol 19 (1) ◽  
pp. 67-76 ◽  
Author(s):  
D. McCarthy ◽  
S.A. Vanstone
Keyword(s):  
Projective Planes ◽  
Finite Projective Planes

scholarly journals Analysis of Some Properties from Construction of Finite Projective Planes of Order 2 and 3

CAUCHY ◽  
2016 ◽  
Vol 4 (3) ◽  
pp. 131
Author(s):  
Vira Hari Krisnawati ◽  
Corina Karim
Keyword(s):  
Automorphism Group ◽  
Projective Plane ◽  
Symmetric Group ◽  
Block Design ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Steiner System ◽  
Incidence Relation ◽  
Finite Projective Planes ◽  
Set Of Points

<p class="abstract"><span lang="IN">In combinatorial mathematics, a Steiner system is a type of block design. Specifically, a Steiner system <em>S</em>(<em>t</em>, <em>k</em>, <em>v</em>) is a set of <em>v</em> points and <em>k</em> blocks which satisfy that every <em>t</em>-subset of <em>v</em>-set of points appear in the unique block. It is well-known that a finite projective plane is one examples of Steiner system with <em>t</em> = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order.</span></p><p class="abstract"><span lang="IN">In this paper, we observe some properties from construction of finite projective planes of order 2 and 3. Also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group.</span></p>

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