Karsten Schölzel

A Classification of Partial Boolean Clones

We study intervals

Relation Graphs and Partial Clones on a 2-Element Set

\mathcal{I}(A)

Countable Intervals of Partial Clones

of partial clones whose total functions constitute a (total) clone A. In the Boolean case, we provide a complete classification of such intervals(according to whether the interval is finite or infinite), and determine the size of each finite interval

The Minimal Covering of Maximal Partial Clones in 4-valued Logic

\mathcal{I}(A)

Hypomorphic Sperner Systems and Non-Reconstructible Functions

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Galois Theory for Partial Clones and Some Relational Clones

In a recent paper, the authors show that the sublattice of partial clones that preserve the relation {(0,0),(0,1),(1,0)} is of continuum cardinality on 2. In this paper we give an alternative proof to this result by making use of a representation of relations derived from {(0,0),(0,1),(1,0)} in terms of certain types of graphs. As a by-product, this tool brings some light into the understanding of the structure of this uncountable sublattice of strong partial clones.

Number of Maximal Partial Clones

Let k ≥ 2 and A be a k-element set. We construct countably infinite unrefinable chains of strong partial clones on A. This provides the first known examples of countably infinite intervals of strong partial clones on a finite set with at least two elements.

A Note on Intervals of Słupecki Partial Clones

A partial function f on a k-element set Ek is a partial Sheffer function if every partial function on Ek is definable in terms of f. Since this holds if and only if f belongs to no maximal partial clone on Ek, a characterization of partial Sheffer functions reduces to finding families of minimal coverings of maximal partial clones on Ek. It is shown that there is only one minimal covering for k = 4 and it is determined. Additionally all 1 102 coherent relations for k = 4 are given in a full list.

On the Nonexistence of Minimal Strong Partial Clones

A reconstruction problem is formulated for Sperner systems, and infinite families of non-reconstructible Sperner systems are presented. This has an application to a reconstruction problem for functions of several arguments and identification minors. Sperner systems being representations of certain monotone functions, infinite families of non-reconstructible functions are thus obtained. The clones of Boolean functions are completely classified in regard to reconstructibility.

Hereditarily Rigid Relations: Dedicated to Professor I.G. Rosenberg on the Occasion of His 80-th Birthday

A Galois connection between partial clones and a new variant of relation algebras is established. We introduce a new elementary operation on relations which captures the difference between total and partial clones and allows us to adapt the proof of the Galois connection from the total case to the partial case. This Galois connection is able to capture all partial clones and is not restricted to strong partial clones as in previous work.