32 research outputs found
Relationally defined clones of tree functions closed under selection or primitive recursion
We investigate classes of tree functions which are closed under composition and primitive recursion or selection (a restricted form of recursion). The main result is the characterization of those finitary relations Ď‚ (on the set of all trees of a fixed signature) for which the clone of tree functions preserving Ď‚ is closed under selection. Moreover, it turns out that such clones are closed also under primitive recursion
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
A General Galois Theory for Operations and Relations in Arbitrary Categories
In this paper, we generalize the notions of polymorphisms and invariant relations to arbitrary categories. This leads us to a Galois connection that coincides with the classical case from universal algebra if the underlying category is the category of sets, but remains applicable no matter how the category is changed. In analogy to the situation in universal algebra, we characterize the Galois closed classes by local closures of clones of operations and local closures of what we will introduce as clones of (generalized) relations. Since the approach is built on purely category-theoretic properties, we will also discuss the dualization of our notions
Taylor's modularity conjecture and related problems for idempotent varieties
We provide a partial result on Taylor's modularity conjecture, and several
related problems. Namely, we show that the interpretability join of two
idempotent varieties that are not congruence modular is not congruence modular
either, and we prove an analogue for idempotent varieties with a cube term.
Also, similar results are proved for linear varieties and the properties of
congruence modularity, having a cube term, congruence -permutability for a
fixed , and satisfying a non-trivial congruence identity.Comment: 27 page