Karsten Schölzel

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The following result has been shown recently in the form of a dichotomy: For every total clone C on 2 := {0, 1}, the set I(C) of all partial clones on 2 whose total component is C, is either finite or of continuum cardinality. In this paper we show that the dichotomy holds, even if only strong partial clones are considered, i.e., partial clones which are(More)
The following natural problem, first considered by D. Lau, has been tackled by several authors recently: Let C be a total clone on 2 := {0, 1}. Describe the interval I(C) of all partial clones on 2 whose total component is C. We establish some results in this direction and combine them with previous ones to show the following dichotomy result: For every(More)
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(More)
The class of threshold functions is known to be characterizable by functional equations or, equivalently, by pairs of relations, which are called relational constraints. It was shown by Hellerstein that this class cannot be characterized by a finite number of such objects. In this paper, we investigate classes of threshold functions which arise as(More)
All maximal partial clones on 4-element, 5-element, and 6-element sets have been found and are compared to the case of maximal clones of all total functions. Due to the large numbers of maximal partial clones other criteria to check for generating systems of all partial functions are analyzed.