On a stronger reconstruction notion for monoids and clones

Abstract Motivated by reconstruction results by Rubin, we introduce a new reconstruction notion for permutation groups, transformation monoids and clones, called automatic action compatibility, which entails automatic homeomorphicity. We further give a characterization of automatic homeomorphicity for transformation monoids on arbitrary carriers with a dense group of invertibles having automatic homeomorphicity. We then show how to lift automatic action compatibility from groups to monoids and from monoids to clones under fairly weak assumptions. We finally employ these theorems to get automatic action compatibility results for monoids and clones over several well-known countable structures, including the strictly ordered rationals, the directed and undirected version of the random graph, the random tournament and bipartite graph, the generic strictly ordered set, and the directed and undirected versions of the universal homogeneous Henson graphs.

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Affine Primitive Groups and Semisymmetric Graphs

The Electronic Journal of Combinatorics ◽

10.37236/2549 ◽

2013 ◽

Vol 20 (2) ◽

Cited By ~ 1

Author(s):

Hua Han ◽

Zaiping Lu

Keyword(s):

Bipartite Graph ◽

Complete Bipartite Graph ◽

Infinite Family ◽

Permutation Groups ◽

Primitive Groups ◽

The Third ◽

Semisymmetric Graphs

In this paper, we investigate semisymmetric graphs which arise from affine primitive permutation groups. We give a characterization of such graphs, and then construct an infinite family of semisymmetric graphs which contains the Gray graph as the third smallest member. Then, as a consequence, we obtain a factorization,of the complete bipartite graph $K_{p^{sp^t},p^{sp^t}}$ into connected semisymmetric graphs, where $p$ is an prime, $1\le t\le s$ with $s\ge2$ while $p=2$.

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Decidability of Definability

Journal of Symbolic Logic ◽

10.2178/jsl.7804020 ◽

2013 ◽

Vol 78 (4) ◽

pp. 1036-1054 ◽

Cited By ~ 16

Author(s):

Manuel Bodirsky ◽

Michael Pinsker ◽

Todor Tsankov

Keyword(s):

Random Graph ◽

Topological Dynamics ◽

Homogeneous Structure ◽

First Order ◽

Computational Problem ◽

Random Tournament ◽

Universal Quantification ◽

Countably Infinite ◽

Existential Quantification

AbstractFor a fixed countably infinite structure Γ with finite relational signature τ, we study the following computational problem: input are quantifier-free τ-formulas ϕ0, ϕ1, …, ϕn that define relations R0, R1, …, Rn over Γ. The question is whether the relation R0 is primitive positive definable from R1, …, Rn, i.e., definable by a first-order formula that uses only relation symbols for R1, …, Rn, equality, conjunctions, and existential quantification (disjunction, negation, and universal quantification are forbidden).We show decidability of this problem for all structures Γ that have a first-order definition in an ordered homogeneous structure Δ with a finite relational signature whose age is a Ramsey class and determined by finitely many forbidden substructures. Examples of structures Γ with this property are the order of the rationals, the random graph, the homogeneous universal poset, the random tournament, all homogeneous universal C-relations, and many more. We also obtain decidability of the problem when we replace primitive positive definability by existential positive, or existential definability. Our proof makes use of universal algebraic and model theoretic concepts, Ramsey theory, and a recent characterization of Ramsey classes in topological dynamics.

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ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP

Nagoya Mathematical Journal ◽

10.1017/nmj.2021.6 ◽

2021 ◽

pp. 1-40

Author(s):

NICK GILL ◽

BIANCA LODÀ ◽

PABLO SPIGA

Keyword(s):

Permutation Group ◽

Model Theory ◽

Independent Set ◽

Maximum Size ◽

Proper Subset ◽

Permutation Groups ◽

Finite Permutation Group ◽

Pointwise Stabilizer ◽

Relational Complexity

Abstract Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

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Face-Width of Pfaffian Braces and Polyhex Graphs on Surfaces

The Electronic Journal of Combinatorics ◽

10.37236/3540 ◽

2014 ◽

Vol 21 (4) ◽

Cited By ~ 1

Author(s):

Dong Ye ◽

Heping Zhang

Keyword(s):

Bipartite Graph ◽

Polynomial Time ◽

Perfect Matching ◽

Cartesian Product ◽

Perfect Matchings ◽

Face Width ◽

Graphs On Surfaces ◽

Positive Genus ◽

Pfaffian Graph

A graph $G$ with a perfect matching is Pfaffian if it admits an orientation $D$ such that every central cycle $C$ (i.e. $C$ is of even size and $G-V(C)$ has a perfect matching) has an odd number of edges oriented in either direction of the cycle. It is known that the number of perfect matchings of a Pfaffian graph can be computed in polynomial time. In this paper, we show that every embedding of a Pfaffian brace (i.e. 2-extendable bipartite graph) on a surface with a positive genus has face-width at most 3. Further, we study Pfaffian cubic braces and obtain a characterization of Pfaffian polyhex graphs: a polyhex graph is Pfaffian if and only if it is either non-bipartite or isomorphic to the cube, or the Heawood graph, or the Cartesian product $C_k\times K_2$ for even integers $k\ge 6$.

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Compatible Tight Riesz Orders

Canadian Journal of Mathematics ◽

10.4153/cjm-1976-024-4 ◽

1976 ◽

Vol 28 (1) ◽

pp. 186-200 ◽

Cited By ~ 2

Author(s):

A. M. W. Glass

Keyword(s):

Geometric Characterization ◽

Permutation Groups ◽

Algebraic Characterization ◽

Partially Ordered ◽

Ordered Field ◽

Ordered Groups

N. R. Reilly has obtained an algebraic characterization of the compatible tight Riesz orders that can be supported by certain partially ordered groups [13; 14]. The purpose of this paper is to give a “geometric“ characterization by the use of ordered permutation groups. Our restrictions on the partially ordered groups will likewise be geometric rather than algebraic. Davis and Bolz [3] have done some work on groups of all order-preserving permutations of a totally ordered field; from our more general theorems, we will be able to recapture their results.

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On the Partially Ordered Set of Prime Ideals of a Distributive Lattice

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-097-3 ◽

1971 ◽

Vol 23 (5) ◽

pp. 866-874 ◽

Cited By ~ 10

Author(s):

Raymond Balbes

Keyword(s):

Distributive Lattice ◽

Partially Ordered Set ◽

Ordered Set ◽

Complete Characterization ◽

Distributive Lattices ◽

Prime Ideals ◽

Partially Ordered ◽

Irreducible Elements

For a distributive lattice L, let denote the poset of all prime ideals of L together with ∅ and L. This paper is concerned with the following type of problem. Given a class of distributive lattices, characterize all posets P for which for some . Such a poset P will be called representable over. For example, if is the class of all relatively complemented distributive lattices, then P is representable over if and only if P is a totally unordered poset with 0, 1 adjoined. One of our main results is a complete characterization of those posets P which are representable over the class of distributive lattices which are generated by their meet irreducible elements. The problem of determining which posets P are representable over the class of all distributive lattices appears to be very difficult.

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MULTIPLE TRANSITIVITY AND MIN-WISE INDEPENDENCE IN PERMUTATION GROUPS

Journal of Algebra and Its Applications ◽

10.1142/s0219498804000952 ◽

2004 ◽

Vol 03 (04) ◽

pp. 427-435

Author(s):

C. FRANCHI

Keyword(s):

Positive Integer ◽

Permutation Group ◽

Ordered Set ◽

Permutation Groups ◽

Linearly Ordered Set

Let Ω be a finite linearly ordered set and let k be a positive integer. A permutation group G on Ω is called co-k-restricted min-wise independent on Ω if [Formula: see text] for any X⊆Ω such that |X|≥|Ω|-k+1 and for any x∈X. We show that co-k-restricted min-wise independent groups are exactly the groups with the property that for each subset X⊆Ω with |X|≤k-1, the stabilizer G{X} of X in G is transitive on Ω\X. Using this fact, we determine all co-k-restricted min-wise independent groups.

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Stratified systems of logic

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100017722 ◽

1943 ◽

Vol 39 (2) ◽

pp. 69-83 ◽

Cited By ~ 9

Author(s):

M. H. A. Newman

Keyword(s):

Mathematical Logic ◽

Partially Ordered Set ◽

General Class ◽

Ordered Set ◽

Partially Ordered ◽

New Foundations

The suffixes used in logic to indicate differences of type may be regarded either as belonging to the formalism itself, or as being part of the machinery for deciding which rows of symbols (without suffixes) are to be admitted as significant. The two different attitudes do not necessarily lead to different formalisms, but when types are regarded as only one way of regulating the calculus it is natural to consider other possible ways, in particular the direct characterization of the significant formulae. Direct criteria for stratification were given by Quine, in his ‘New Foundations for Mathematical Logic’ (7). In the corresponding typed form of this theory ordinary integers are adequate as type-suffixes, and the direct description is correspondingly simple, but in other theories, including that recently proposed by Church(4), a partially ordered set of types must be used. In the present paper criteria, equivalent to the existence of a correct typing, are given for a general class of formalisms, which includes Church's system, several systems proposed by Quine, and (with some slight modifications, given in the last paragraph) Principia Mathematica. (The discussion has been given this general form rather with a view to clarity than to comprehensiveness.)

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