158 research outputs found
A counterexample to the reconstruction of -categorical structures from their endomorphism monoids
We present an example of two countable -categorical structures, one
of which has a finite relational language, whose endomorphism monoids are
isomorphic as abstract monoids, but not as topological monoids -- in other
words, no isomorphism between these monoids is a homeomorphism. For the same
two structures, the automorphism groups and polymorphism clones are isomorphic,
but not topologically isomorphic. In particular, there exists a countable
-categorical structure in a finite relational language which can
neither be reconstructed up to first-order bi-interpretations from its
automorphism group, nor up to existential positive bi-interpretations from its
endomorphism monoid, nor up to primitive positive bi-interpretations from its
polymorphism clone.Comment: 17 page
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
Clones from Creatures
A clone on a set X is a set of finitary operations on X which contains all
the projections and is closed under composition.
The set of all clones forms a complete lattice Cl(X) with greatest element O,
the set of all finitary operations. For finite sets X the lattice is "dually
atomic": every clone other than O is below a coatom of Cl(X).
It was open whether Cl(X) is also dually atomic for infinite X. Assuming the
continuum hypothesis, we show that there is a clone C on a countable set such
that the interval of clones above C is linearly ordered, uncountable, and has
no coatoms.Comment: LaTeX2e, 20 pages. Revised version: some concepts simplified, proof
details adde
A survey of clones on infinite sets
A clone on a set X is a set of finitary operations on X which contains all
projections and which is moreover closed under functional composition. Ordering
all clones on X by inclusion, one obtains a complete algebraic lattice, called
the clone lattice. We summarize what we know about the clone lattice on an
infinite base set X and formulate what we consider the most important open
problems.Comment: 37 page
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