30 research outputs found
On a stronger reconstruction notion for monoids and clones
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.Comment: 29 pp; Changes v1-->v2::typos corr.|L3.5+pf extended|Rem3.7 added|C.
Pech found out that arg of L5.3-v1 solved Probl2-v1|L5.3, C5.4, Probl2 of v1
removed|C5.2, R5.4 new, contain parts of pf of L5.3-v1|L5.2-v1 is now
L5.3,merged with concl of C5.4-v1,L5.3-v2 extends C5.4-v1|abstract, intro
updated|ref[24] added|part of L5.3-v1 is L2.1(e)-v2, another part merged with
pf of L5.2-v1 => L5.3-v
Relational Structure Theory: A Localisation Theory for Algebraic Structures
This thesis extends a localisation theory for finite algebras to certain classes of infinite structures. Based on ideas and constructions originally stemming from Tame Congruence Theory, algebras are studied via local restrictions of their relational counterpart (Relational Structure Theory). In this respect, first those subsets are identified that are suitable for such a localisation process, i. e. that are compatible with the relational clone structure of the counterpart of an algebra. It is then studied which properties of the global algebra can be transferred to its localisations, called neighbourhoods. Thereafter, it is discussed how this process can be reversed, leading to the concept of covers. These are collections of neighbourhoods that allow information retrieval about the global structure from knowledge about the local restrictions. Subsequently, covers are characterised in terms of a decomposition equation, and connections to categorical equivalences of algebras are explored. In the second half of the thesis, a refinement concept for covers is introduced in order to find optimal, non-refinable covers, eventually leading to practical algorithms for their determination. Finally, the text establishes further theoretical foundations, e. g. several irreducibility notions, in order to ensure existence of non-refinable covers via an intrinsic characterisation, and to prove under some conditions that they are uniquely determined in a canonical sense. At last, the applicability of the developed techniques is demonstrated using two clear expository examples.:1 Introduction
2 Preliminaries and Notation
2.1 Functions, operations and relations
2.2 Algebras and relational structures
2.3 Clones
3 Relational Structure Theory
3.1 Finding suitable subsets for localisation
3.2 Neighbourhoods
3.3 The restricted algebra A|U
3.4 Covers
3.5 Refinement
3.6 Irreducibility notions
3.7 Intrinsic description of non-refinable covers
3.8 Elaborated example
4 Problems and Prospects for Future Research
Acknowledgements
Index of Notation
Index of Terms
BibliographyDiese Dissertation erweitert eine Lokalisierungstheorie fĂŒr endliche Algebren auf gewisse Klassen unendlicher Strukturen. Basierend auf Ideen und Konstruktionen, die ursprĂŒnglich der Tame Congruence Theory entstammen, werden Algebren ĂŒber lokale EinschrĂ€nkungen ihres relationalen GegenstĂŒcks untersucht (Relationale Strukturtheorie). In diesem Zusammenhang werden zunĂ€chst diejenigen Teilmengen identifiziert, welche fĂŒr einen solchen LokalisierungsprozeĂ geeignet sind, d. h., die mit der Relationenklonstruktur auf dem GegenstĂŒck einer Algebra kompatibel sind. Es wird dann untersucht, welche Eigenschaften der globalen Algebra auf ihre Lokalisierungen, genannt Umgebungen, ĂŒbertragen werden können. Nachfolgend wird diskutiert, wie dieser Vorgang umgekehrt werden kann, was zum Begriff der Ăberdeckungen fĂŒhrt. Dies sind Systeme von Umgebungen, welche die RĂŒckgewinnung von Informationen ĂŒber die globale Struktur aus Kenntnis ihrer lokalen EinschrĂ€nkungen erlauben. Sodann werden Ăberdeckungen durch eine Zerlegungsgleichung charakterisiert und BezĂŒge zu kategoriellen Ăquivalenzen von Algebren hergestellt. In der zweiten HĂ€lfte der Arbeit wird ein Verfeinerungsbegriff fĂŒr Ăberdeckungen eingefĂŒhrt, um optimale, nichtverfeinerbare Ăberdeckungen zu finden, was letztlich zu praktischen Algorithmen zu ihrer Bestimmung fĂŒhrt. SchlieĂlich erarbeitet der Text weitere theoretische Grundlagen, beispielsweise mehrere IrreduzibilitĂ€tsbegriffe, um die Existenz nichtverfeinerbarer Ăberdeckungen vermöge einer intrinsischen Charakterisierung sicherzustellen und, unter gewissen Bedingungen, zu beweisen, daĂ sie in kanonischer Weise eindeutig bestimmt sind. SchluĂendlich wird die Anwendbarkeit der entwickelten Methoden an zwei ĂŒbersichtlichen Beispielen demonstriert.:1 Introduction
2 Preliminaries and Notation
2.1 Functions, operations and relations
2.2 Algebras and relational structures
2.3 Clones
3 Relational Structure Theory
3.1 Finding suitable subsets for localisation
3.2 Neighbourhoods
3.3 The restricted algebra A|U
3.4 Covers
3.5 Refinement
3.6 Irreducibility notions
3.7 Intrinsic description of non-refinable covers
3.8 Elaborated example
4 Problems and Prospects for Future Research
Acknowledgements
Index of Notation
Index of Terms
Bibliograph
The number of clones determined by disjunctions of unary relations
We consider finitary relations (also known as crosses) that are definable via
finite disjunctions of unary relations, i.e. subsets, taken from a fixed finite
parameter set . We prove that whenever contains at least one
non-empty relation distinct from the full carrier set, there is a countably
infinite number of polymorphism clones determined by relations that are
disjunctively definable from . Finally, we extend our result to
finitely related polymorphism clones and countably infinite sets .Comment: manuscript to be published in Theory of Computing System
Minimal clones with many majority operations
We present two minimal clones containing 26 and 78 majority operations
respectively, more than any other previously known example.Comment: 11 page
Some Examples of Minimal Groupoids on a Finite Set (Algebraic System, Logic, Language and Related Areas in Computer Science)
A minimal clone is an atom in the lattice of clones. The classification of minimal clones on a finite set still remains unsolved. A minimal groupoid is a minimal clone generated by a binary idempotent function. In this paper we report some examples of minimal groupoids generated by binary functions which resemble projections
On when the union of two algebraic sets is algebraic
In universal algebraic geometry, an algebra is called an equational domain if
the union of two algebraic sets is algebraic. We characterize equational
domains, with respect to polynomial equations, inside congruence permutable
varieties, and with respect to term equations, among all algebras of size two
and all algebras of size three with a cyclic automorphism. Furthermore, for
each size at least three, we prove that, modulo term equivalence, there is a
continuum of equational domains of that size.Comment: 50 pages, 1 figure, 1 tabl
On the relationship of maximal C-clones and maximal clones
A restricted version of the Galois connection between polymorphisms and invariants, called PolâCInv, is studied, where the invariant relations are restricted to so-called clausal relations. In this context, the relationship of maximal C-clones and maximal clones is investigated. It is shown that, with the exception of one special case occurring for Boolean domains, maximal C-clones are never maximal clones.:1 Introduction
2 Preliminaries
3 Proof of the main theorem
3.1 Principle of proof
3.2 Bounded orders
3.3 Non-trivial congruences
3.4 Selfdual functions
3.5 Quasilinear functions
3.6 Functions preserving central and h-regular relations
4 Concluding remarks
ReferencesWir untersuchen eine eingeschrĂ€nkte Variante der Galoisverbindung zwischen Polymorphismen und invarianten Relationen, bezeichnet mit PolâCInv, wobei die invarianten Relationen auf sogenannte klausale Relationen beschrĂ€nkt werden. In diesem Zusammenhang wird die Beziehung zwischen maximalen C-Klonen und maximalen Klonen betrachtet. Es wird gezeigt, daĂ, mit Ausnahme eines Spezialfalles fĂŒr Boolesche Grundmengen, maximale C-Klone niemals maximale Klone sind.:1 Introduction
2 Preliminaries
3 Proof of the main theorem
3.1 Principle of proof
3.2 Bounded orders
3.3 Non-trivial congruences
3.4 Selfdual functions
3.5 Quasilinear functions
3.6 Functions preserving central and h-regular relations
4 Concluding remarks
Reference
On the relationship of maximal C-clones and maximal clones
A restricted version of the Galois connection between polymorphisms and invariants, called PolâCInv, is studied, where the invariant relations are restricted to so-called clausal relations. In this context, the relationship of maximal C-clones and maximal clones is investigated. It is shown that, with the exception of one special case occurring for Boolean domains, maximal C-clones are never maximal clones.:1 Introduction
2 Preliminaries
3 Proof of the main theorem
3.1 Principle of proof
3.2 Bounded orders
3.3 Non-trivial congruences
3.4 Selfdual functions
3.5 Quasilinear functions
3.6 Functions preserving central and h-regular relations
4 Concluding remarks
ReferencesWir untersuchen eine eingeschrĂ€nkte Variante der Galoisverbindung zwischen Polymorphismen und invarianten Relationen, bezeichnet mit PolâCInv, wobei die invarianten Relationen auf sogenannte klausale Relationen beschrĂ€nkt werden. In diesem Zusammenhang wird die Beziehung zwischen maximalen C-Klonen und maximalen Klonen betrachtet. Es wird gezeigt, daĂ, mit Ausnahme eines Spezialfalles fĂŒr Boolesche Grundmengen, maximale C-Klone niemals maximale Klone sind.:1 Introduction
2 Preliminaries
3 Proof of the main theorem
3.1 Principle of proof
3.2 Bounded orders
3.3 Non-trivial congruences
3.4 Selfdual functions
3.5 Quasilinear functions
3.6 Functions preserving central and h-regular relations
4 Concluding remarks
Reference