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See detailRelation graphs and partial clones on a 2-element set.
Schölzel, Karsten UL; Couceiro, Miguel; Haddad, Lucien et al

in Multiple-Valued Logic (ISMVL), 2014 IEEE 44rd International Symposium on (2014)

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 ... [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 brings some light into the understanding of the structure of this uncountable sublattice of strong partial clones. [less ▲]

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See detailPartial R-clones
Schölzel, Karsten UL; Romov, Boris

in Multiple-Valued Logic (ISMVL), 2014 IEEE 44rd International Symposium on (2014)

In [Romov, ISMVL 2013] the first author introduced a type of partial clones as the intersection of some special infinite descending chains similar to I. Rosenberg (1972) in the case of total clones. We ... [more ▼]

In [Romov, ISMVL 2013] the first author introduced a type of partial clones as the intersection of some special infinite descending chains similar to I. Rosenberg (1972) in the case of total clones. We provide a characterization of these clones in terms of their invariants, and propose a generalization of such clones, which we will call partial R-clones. We show that there are only finitely many partial R-clones. Furthermore, we investigate some properties related to their position in the lattice of partial clones. [less ▲]

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See detailCountable intervals of partial clones
Haddad, Lucien; Schölzel, Karsten UL

in Multiple-Valued Logic (ISMVL), 2014 IEEE 44rd International Symposium on (2014)

Let $k \ge 2$ 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 ... [more ▼]

Let $k \ge 2$ 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. [less ▲]

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