Inequivalent representations of geometric relation algebras

2003 ◽  
Vol 68 (1) ◽  
pp. 267-310 ◽  
Author(s):  
Steven Givant
Keyword(s):  
Automorphism Group ◽  
Collineation Group ◽  
Relation Algebra ◽  
Projective Line ◽  
Finite Geometries ◽  
Relation Algebras ◽  
Geometric Relation ◽  
Image Position ◽  
Projective Lines ◽  
Inequivalent Representations

AbstractIt is shown that the automorphism group of a relation algebra constructed from a projective geometry P is isomorphic to the collineation group of P. Also, the base automorphism group of a representation of over an affine geometry D is isomorphic to the quotient of the collineation group of D by the dilatation subgroup. Consequently, the total number of inequivalent representations of , for finite geometries P, is the sum of the numberswhere D ranges over a list of the non-isomorphic affine geometries having P as their geometry at infinity. This formula is used to compute the number of inequivalent representations of relation algebras constructed over projective lines of order at most 10. For instance, the relation algebra constructed over the projective line of order 9 has 56,700 mutually inequivalent representations.

scholarly journals THE VARIETY OF COSET RELATION ALGEBRAS

2018 ◽  
Vol 83 (04) ◽  
pp. 1595-1609 ◽  
Author(s):  
STEVEN GIVANT ◽  
HAJNAL ANDRÉKA
Keyword(s):  
Large Class ◽  
Identity Element ◽  
Relation Algebra ◽  
Order Theory ◽  
Relation Algebras ◽  
First Order ◽  
First Order Theory ◽  
Atomic Pair ◽  
Finite Set ◽  
Image Position

AbstractGivant [6] generalized the notion of an atomic pair-dense relation algebra from Maddux [13] by defining the notion of a measurable relation algebra, that is to say, a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the “size” of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). In Andréka--Givant [2], a large class of examples of such algebras is constructed from systems of groups, coordinated systems of isomorphisms between quotients of the groups, and systems of cosets that are used to “shift” the operation of relative multiplication. In Givant--Andréka [8], it is shown that the class of these full coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic and complete measurable relation algebra is isomorphic to a full coset relation algebra.Call an algebra $\mathfrak{A}$ a coset relation algebra if $\mathfrak{A}$ is embeddable into some full coset relation algebra. In the present article, it is shown that the class of coset relation algebras is equationally axiomatizable (that is to say, it is a variety), but that no finite set of sentences suffices to axiomatize the class (that is to say, the class is not finitely axiomatizable).

Relation algebra reducts of cylindric algebras and complete representations

2007 ◽  
Vol 72 (2) ◽  
pp. 673-703 ◽  
Author(s):  
Robin Hirsch
Keyword(s):  
Elementary Theory ◽  
Winning Strategy ◽  
Relation Algebra ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
Complete Representation ◽  
Recursively Enumerable ◽  
Complete Representations ◽  
Atomic Relation ◽  
Image Position

AbstractWe show, for any ordinal γ ≥ 3, that the class ℜaCAγ is pseudo-elementary and has a recursively enumerable elementary theory. ScK denotes the class of strong subalgebras of members of the class K. We devise games, Fn (3 ≤ n ≤ ω), G, H, and show, for an atomic relation algebra with countably many atoms, thatfor 3 ≤ n < ω. We use these games to show, for γ > 5 and any class K of relation algebras satisfyingthat K is not closed under subalgebras and is not elementary. For infinite γ, the inclusion ℜaCAγ ⊂ ScℜaCAγ is strict.For infinite γ and for a countable relation algebra we show that has a complete representation if and only if is atomic and ∃ has a winning strategy in F (At()) if and only if is atomic and ∈ ScℜaCAγ.

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.

-SPHERICAL FUNCTIONS ON ABELIAN SEMIGROUPS

2017 ◽  
Vol 96 (3) ◽  
pp. 479-486 ◽  
Author(s):  
RADOSŁAW ŁUKASIK
Keyword(s):  
Functional Equation ◽  
Automorphism Group ◽  
Spherical Functions ◽  
Abelian Semigroup ◽  
Image Position

We present the form of the solutions $f:S\rightarrow \mathbb{C}$ of the functional equation $$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D706}\in K}f(x+\unicode[STIX]{x1D706}y)=|K|f(x)f(y)\quad \text{for }x,y\in S,\end{eqnarray}$$ where $f$ satisfies the condition $f(\sum _{\unicode[STIX]{x1D706}\in K}\unicode[STIX]{x1D706}x)\neq 0$ for all $x\in S$, $(S,+)$ is an abelian semigroup and $K$ is a subgroup of the automorphism group of $S$.

The Maximum Order of the Group of a Tournament

1967 ◽  
Vol 10 (4) ◽  
pp. 503-505 ◽  
Author(s):  
John D. Dixon
Keyword(s):  
Automorphism Group ◽  
Maximum Order ◽  
Image Position

To each tournament Tn with n nodes n there corresponds the automorphism group G(Tn) consisting n of all dominance preserving permutations of the set of nodes. In a recent paper [3], Myron Goldberg and J. W. Moon consider the maximum order g(n) which the group of a tournament with n nodes may have. Among other results they prove that12

Non-finite-axiomatizability results in algebraic logic

1992 ◽  
Vol 57 (3) ◽  
pp. 832-843 ◽  
Author(s):  
Balázs Biró
Keyword(s):  
Algebraic Logic ◽  
Relation Algebra ◽  
Equational Theory ◽  
Negative Answer ◽  
Relation Algebras ◽  
Representable Relation Algebra ◽  
Finite Set ◽  
Variable Symbol ◽  
Axiom Systems ◽  
Representable Relation

This paper deals with relation, cylindric and polyadic equality algebras. First of all it addresses a problem of B. Jónsson. He asked whether relation set algebras can be expanded by finitely many new operations in a “reasonable” way so that the class of these expansions would possess a finite equational base. The present paper gives a negative answer to this problem: Our main theorem states that whenever Rs+ is a class that consists of expansions of relation set algebras such that each operation of Rs+ is logical in Jónsson's sense, i.e., is the algebraic counterpart of some (derived) connective of first-order logic, then the equational theory of Rs+ has no finite axiom systems. Similar results are stated for the other classes mentioned above. As a corollary to this theorem we can solve a problem of Tarski and Givant [87], Namely, we claim that the valid formulas of certain languages cannot be axiomatized by a finite set of logical axiom schemes. At the same time we give a negative solution for a version of a problem of Henkin and Monk [74] (cf. also Monk [70] and Németi [89]).Throughout we use the terminology, notation and results of Henkin, Monk, Tarski [71] and [85]. We also use results of Maddux [89a].Notation. RA denotes the class of relation algebras, Rs denotes the class of relation set algebras and RRA is the class of representable relation algebras, i.e. the class of subdirect products of relation set algebras. The symbols RA, Rs and RRA abbreviate also the expressions relation algebra, relation set algebra and representable relation algebra, respectively.For any class C of similar algebras EqC is the set of identities that hold in C, while Eq1C is the set of those identities in EqC that contain at most one variable symbol. (We note that Henkin et al. [85] uses the symbol EqC in another sense.)

Rigid homogeneous chains

1981 ◽  
Vol 89 (1) ◽  
pp. 07-17 ◽  
Author(s):  
A. M. W. Glass ◽  
Yuri Gurevich ◽  
W. Charles Holland ◽  
Saharon Shelah
Keyword(s):  
Automorphism Group ◽  
General Problem ◽  
Automorphism Groups ◽  
Ordered Sets ◽  
First Order ◽  
Image Position

Classifying (unordered) sets by the elementary (first order) properties of their automorphism groups was undertaken in (7), (9) and (11). For example, if Ω is a set whose automorphism group, S(Ω), satisfiesthen Ω has cardinality at most ℵ0 and conversely (see (7)). We are interested in classifying homogeneous totally ordered sets (homogeneous chains, for short) by the elementary properties of their automorphism groups. (Note that we use ‘homogeneous’ here to mean that the automorphism group is transitive.) This study was begun in (4) and (5). For any set Ω, S(Ω) is primitive (i.e. has no congruences). However, the automorphism group of a homogeneous chain need not be o-primitive (i.e. it may have convex congruences). Fortunately, ‘o-primitive’ is a property that can be captured by a first order sentence for automorphisms of homogeneous chains. Hence our general problem falls naturally into two parts. The first is to classify (first order) the homogeneous chains whose automorphism groups are o-primitive; the second is to determine how the o-primitive components are related for arbitrary homogeneous chains whose automorphism groups are elementarily equivalent.

ℵ0-categorical structures with arbitrarily fast growth of algebraic closure

2002 ◽  
Vol 67 (3) ◽  
pp. 897-909
Author(s):  
David M. Evans ◽  
M. E. Pantano
Keyword(s):  
Growth Rate ◽  
Automorphism Group ◽  
Finite Subset ◽  
Growth Rates ◽  
Algebraic Closure ◽  
Fast Growth ◽  
Permutation Groups ◽  
Natural Numbers ◽  
Categorical Structure ◽  
Image Position

Various results have been proved about growth rates of certain sequences of integers associated with infinite permutation groups. Most of these concern the number of orbits of the automorphism group of an ℵ0-categorical structure on the set of unordered n-subsets or on the set of n-tuples of elements of . (Recall that by the Ryll-Nardzewski Theorem, if is countable and ℵ0-categorical, the number of the orbits of its automorphism group Aut() on the set of n-tuples from is finite and equals the number of complete n-types consistent with the theory of .) The book [Ca90] is a convenient reference for these results. One of the oldest (in the realms of ‘folklore’) is that for any sequence (Kn)n∈ℕ of natural numbers there is a countable ℵ0-categorical structure such that the number of orbits of Aut() on the set of n-tuples from is greater than kn for all n.These investigations suggested the study of the growth rate of another sequence. Let be an ℵ0-categorical structure and X be a finite subset of . Let acl(X) be the algebraic closure of X, that is, the union of the finite X-definable subsets of . Equivalently, this is the union of the finite orbits on of Aut()(X), the pointwise stabiliser of X in Aut(). Define

scholarly journals On the power of a prime dividing the order of the automorphism group of a finite group

1960 ◽  
Vol 4 (4) ◽  
pp. 163-170 ◽  
Author(s):  
J. C. Howarth
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Image Position ◽  
A Finite Group

The existence of a function g of hhaving the property that pr divides the order of the automorphism group of a finite group G whenever pg divides the order of G was first established by Ledermann and Neumann [4], who showed that the least such function g(h) satisfies the inequalityLater Green [2] improved this estimate toIn the Present paper this will be revised, for sufficiently large h, to

Relation algebra reducts of cylindric algebras and an application to proof theory

2002 ◽  
Vol 67 (1) ◽  
pp. 197-213 ◽  
Author(s):  
Robin Hirsch ◽  
Ian Hodkinson ◽  
Roger D. Maddux
Keyword(s):  
Proof Theory ◽  
Predicate Logic ◽  
Algebraic Logic ◽  
Relation Algebra ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
First Order ◽  
Order Language ◽  
Binary Relation Symbol ◽  
Made In

AbstractWe confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal α ≥ 3 there is a logically valid sentence X, in a first-order language ℒ with equality and exactly one nonlogical binary relation symbol E, such that X contains only 3 variables (each of which may occur arbitrarily many times), X has a proof containing exactly α + 1 variables, but X has no proof containing only α variables. This solves a problem posed by Tarski and Givant in 1987.

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