References of "Haddad, Lucien"
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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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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 detailIntersections with Slupecki Partial Clones on a Finite Set
Schölzel, Karsten UL; Haddad, Lucien

in Multiple-Valued Logic (ISMVL), 2013 IEEE 43rd International Symposium on (2013, May)

We study intersections of partial clones on the k-element set with k ≥ 2. More precisely we consider intersections of Slupecki partial clones with non-Slupecki maximal partial clones on k.

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See detailA Solution to a Problem of D. Lau: Complete Classification of Intervals in the Lattice of Partial Boolean Clones
Schölzel, Karsten UL; Couceiro, Miguel UL; Haddad, Lucien et al

in Multiple-Valued Logic (ISMVL), 2013 IEEE 43rd International Symposium on (2013)

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 ... [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 total clone C on 2, the set I(C) is either finite or of continuum cardinality. [less ▲]

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See detailIntersections of Finitely Generated Maximal Partial Clones
Couceiro, Miguel UL; Haddad, Lucien

in Journal of Multiple-Valued Logic & Soft Computing (2012), 19(1-3), 85-94

Let A be a finite non-singleton set. For A ={0, 1} we show that the set of all self-dual monotonic partial functions is a not finitely generated partial clone on {0, 1} and that it contains a family of ... [more ▼]

Let A be a finite non-singleton set. For A ={0, 1} we show that the set of all self-dual monotonic partial functions is a not finitely generated partial clone on {0, 1} and that it contains a family of partial subclones of continuum cardinality. Moreover, for |A| ≥ 3, we show that there are pairs of finitely generated maximal partial clones whose intersection is a not finitely generated partial clone on A. [less ▲]

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