Lucien Haddad - Academia.edu (original) (raw)

Papers by Lucien Haddad

Journal of Combinatorial Theory, Series A, May 1, 1997

and v 15, a 3-chromatic Steiner triple system of order v all of whose 3-colorings are equitable.

Journal of Multiple-valued Logic and Soft Computing, 2017

Let k be a k-element set. We show that the lattice of all strong partial clones on k has no minim... more Let k be a k-element set. We show that the lattice of all strong partial clones on k has no minimal elements. Moreover, we show that if C is a strong partial clone, then the family of all partial subclones of C is of continuum cardinality. Finally we show that every non-trivial strong partial clone contains a family of continuum cardinality of strong partial subclones.

Let k/spl ges/2, k be a k-element set, /spl rho//sub 1/ and /spl rho//sub 2/ two relations on k a... more Let k/spl ges/2, k be a k-element set, /spl rho//sub 1/ and /spl rho//sub 2/ two relations on k and let /spl rho//sub 1//spl ominus//spl rho//sub 2/ be the concatenation of /spl rho//sub 1/ and /spl rho//sub 2/. We study the link between the partial clones pPol /spl rho//sub 1//spl cap/pPol /spl rho//sub 2/ and pPol (/spl rho//sub 1//spl ominus//spl rho//sub 2/). Using results arising from this study we address the following problem: given two maximal partial clones M/sub 1/ and M/sub 2/ over k, under what conditions is the partial clone M/sub 1//spl cap/M/sub 2/ covered by M/sub 1/ or by M/sub 1/? So far the research in this direction was focused on partial clones of Boolean functions and on Slupecki type maximal partial clones.

The purpose of this note is to present some of our recent results on clones and partial clones. L... more The purpose of this note is to present some of our recent results on clones and partial clones. Let A be a non-singleton finite set and M be a maximal clone on A. If M is determined by a prime affine or an h-universal relation on A, then we show that M is contained in a family of partial clones on A of continuum cardinality.

Let A be a finite non-singleton set. For |A| = 2 we show that the partial clone consisting of all... more Let A be a finite non-singleton set. For |A| = 2 we show that the partial clone consisting of all selfdual monotone partial functions on A is not finitely generated, while it is the intersection of two finitely generated maximal partial clones on A. Moreover for |A| ≥ 3 we show that there are pairs of finitely generated maximal partial clones whose intersection is a non-finitely generated partial clone on A.

Let A be a nonsingleton finite set and M be a family of maximal partial clones with trivial inter... more Let A be a nonsingleton finite set and M be a family of maximal partial clones with trivial intersection over A. What is the smallest possible cardinality of M? Dually, if F is a family of minimal partial clones whose join is the set of all partial functions on A, then what is the smallest possible cardinality of F? The

The following natural problem, first considered by D. Lau, has been tackled by several authors re... 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.

Journal of Automata, Languages and Combinatorics, Jul 1, 2001

The following two problems are addressed in this paper. Let kgeq2k \geq 2kgeq2, k\kk be a kkk-element set... more The following two problems are addressed in this paper. Let kgeq2k \geq 2kgeq2, k\kk be a kkk-element set and MMM be a family of maximal partial clones with trivial intersection over kkk. What is the smallest possible cardinality of MMM? Dually, if FFF is a family of minimal partial clones whose join is the set of all partial functions on kkk, then what is the smallest possible cardinality of FFF? We show that the answer to these two problems is three.

Discussiones Mathematicae Graph Theory, 2004

The Erdős-Faber-Lovász conjecture states that if a graph G is the union of n cliques of size n no... more The Erdős-Faber-Lovász conjecture states that if a graph G is the union of n cliques of size n no two of which share more than one vertex, then χ(G) = n. We provide a formulation of this conjecture in terms of maximal partial clones of partial operations on a set.

We present some of our recent results on partial clones. Let A be a non singleton finite set. For... more We present some of our recent results on partial clones. Let A be a non singleton finite set. For every maximal clone C on A, we find the maximal partial clone on A that contains C. We also construct families of finitely generated maximal partial clones as well as a family of not finitely generated maximal partial clones on A. Furthermore, we study the pairwise intersections of all maximal partial clones of Slupecki type on A.

arXiv (Cornell University), Apr 1, 2021

We continue the investigation of systems of hereditarily rigid relations started in Couceiro, Had... more We continue the investigation of systems of hereditarily rigid relations started in Couceiro, Haddad, Pouzet and Schölzel [1]. We observe that on a set V with m elements, there is a hereditarily rigid set R made of n tournaments if and only if m(m − 1) ≤ 2 n. We ask if the same inequality holds when the tournaments are replaced by linear orders. This problem has an equivalent formulation in terms of separation of linear orders. Let h Lin (m) be the least cardinal n such that there is a family R of n linear orders on an m-element set V such that any two distinct ordered pairs of distinct elements of V are separated by some member of R, then ⌈log 2 (m(m − 1))⌉ ≤ h Lin (m) with equality if m ≤ 7. We ask whether the equality holds for every m. We prove that h Lin (m+1) ≤ h Lin (m)+1. If V is infinite, we show that h Lin (m) = ℵ0 for m ≤ 2 ℵ 0. More generally, we prove that the two equalities h Lin (m) = log2(m) = d(Lin(V)) hold, where log 2 (m) is the least cardinal µ such that m ≤ 2 µ , and d(Lin(V)) is the topological density of the set Lin(V) of linear orders on V (viewed as a subset of the power set P(V × V) equipped with the product topology). These equalities follow from the Generalized Continuum Hypothesis, but we do not know whether they hold without any set theoretical hypothesis.

Bulletin of The Australian Mathematical Society, Oct 1, 1995

Bulletin of The Australian Mathematical Society, Dec 1, 1991

Let A be a finite set. A partial clone on A is a composition closed set of operations containing ... more Let A be a finite set. A partial clone on A is a composition closed set of operations containing all projections. It is well known that the partial clones on A, ordered by inclusion, form a lattice. We show that the minimal partial clones on A are: (a) minimal clones of full operations or (b) generated by partial projections defined on a totally reflexive and totally symmetric domain.

Let k ≥ 2 and let k be a k-element set. A partial clone on k is said to be of Slupecki type if it... more Let k ≥ 2 and let k be a k-element set. A partial clone on k is said to be of Slupecki type if it contains all unary functions on k. We present some of our latest results about intervals of Slupecki type partial clones on k.

Proceedings - International Symposium on Multiple-Valued Logic, May 1, 2007

partial function f on a k-element set k is a partial Sheffer function if every partial function o... more partial function f on a k-element set k is a partial Sheffer function if every partial function on k is definable in terms of f. Since this holds if and only if f belongs to no maximal partial clone on k, a characterization of partial Sheffer functions reduces to finding families of minimal coverings of maximal partial clones on k. We present minimal coverings of maximal partial clones on k for k = 2 and k = 3 and deduce criteria for partial Sheffer functions on a 2-element and a 3-element set.

Canadian Journal of Mathematics, Oct 1, 1994

Let A be a finite set with \A\ > 2. We describe all clones on A containing the set S A of all per... more Let A be a finite set with \A\ > 2. We describe all clones on A containing the set S A of all permutations of A among its unary operations. (A clone on A is a composition closed set of finitary operations on A containing all projections). With a few exceptions such a clone C is either essentially unary or cellular i.e. there exists a monoid M of self-maps of A containing S4 such that either C = M (= all essentially unary operations agreeing with some/ 6 M) or C = ML) T^ where 1 < h < \A\ and Tfj consists of all finitary operations on A taking at most h values. The exceptions are subclones of Burle's clone or of its variant (provided \A\ is even).

HAL (Le Centre pour la Communication Scientifique Directe), 2012

Let A be a finite non-singleton set. For A = {0, 1} we show that the set of all self-dual monoton... 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.

Journal of Combinatorial Theory, Series A, Feb 1, 1996

Using an embedding result for pairwise balanced designs, and colourings of small systems, triplin... more Using an embedding result for pairwise balanced designs, and colourings of small systems, tripling constructions are used to produce equitably eoloured Steiner triple systems. It is shown that when the order is v is large enough with respect to the number r of colours, and v= 1, 3 (mod6), an equitably r-coloured r-chromatic Steiner triple system of order v exists.

Journal of Combinatorial Designs, 1999

Geometric properties are used to determine the chromatic number of AG(4, 3) and to derive some im... more Geometric properties are used to determine the chromatic number of AG(4, 3) and to derive some important facts on the chromatic number of PG(n, 2). It is also shown that a 4-chromatic STS(v) exists for every admissible order v ≥ 21.