6,484 research outputs found
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
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
Equivalence of operations with respect to discriminator clones
For each clone C on a set A there is an associated equivalence relation,
called C-equivalence, on the set of all operations on A, which relates two
operations iff each one is a substitution instance of the other using
operations from C. In this paper we prove that if C is a discriminator clone on
a finite set, then there are only finitely many C-equivalence classes.
Moreover, we show that the smallest discriminator clone is minimal with respect
to this finiteness property. For discriminator clones of Boolean functions we
explicitly describe the associated equivalence relations.Comment: 17 page
The wonderland of reflections
A fundamental fact for the algebraic theory of constraint satisfaction
problems (CSPs) over a fixed template is that pp-interpretations between at
most countable \omega-categorical relational structures have two algebraic
counterparts for their polymorphism clones: a semantic one via the standard
algebraic operators H, S, P, and a syntactic one via clone homomorphisms
(capturing identities). We provide a similar characterization which
incorporates all relational constructions relevant for CSPs, that is,
homomorphic equivalence and adding singletons to cores in addition to
pp-interpretations. For the semantic part we introduce a new construction,
called reflection, and for the syntactic part we find an appropriate weakening
of clone homomorphisms, called h1 clone homomorphisms (capturing identities of
height 1).
As a consequence, the complexity of the CSP of an at most countable
-categorical structure depends only on the identities of height 1
satisfied in its polymorphism clone as well as the the natural uniformity
thereon. This allows us in turn to formulate a new elegant dichotomy conjecture
for the CSPs of reducts of finitely bounded homogeneous structures.
Finally, we reveal a close connection between h1 clone homomorphisms and the
notion of compatibility with projections used in the study of the lattice of
interpretability types of varieties.Comment: 24 page
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