Lucien Haddad

Binary codes and caps

The connection between maximal caps (sometimes called complete caps) and certain binary codes called quasi-perfect codes is described. We provide a geometric approach to the foundational work of Davydov and Tombak who have obtained the exact possible sizes of large maximal caps. A new self-contained proof of the existence and the structure of the largest maximal nonaffine cap in ℙG(n, 2) is given. Combinatorial and geometric consequences are briefly sketched. Some of these, such as the connection with families of symmetric-difference free subsets of a finite set will be developed elsewhere.

On intervals of partial clones of Boolean partial functions

We describe the interval of all partial clones that contain all monotonic idempotent Boolean partial functions as well as the interval of all partial clones that contain all idempotent self-dual Boolean partial functions.

Partial clones containing all permutations

Let A be a finite set. A clone on A is a composition closed set of operations onA containing all the projections. If in this definiton we replace operations by partialoperations, then we obtain a partial clone (this and other concepts will be definedprecisely in Section 2). The full description of all clones containing all the permutationson A among their unary operations is given in [5]. In particular, it is shown that thereare only finitely many such clones. In this paper, we show that this does not holdfor partial clones. Actually, the set of such partial clones is of continuum cardinalityeven for \A\ = 2, in contrast to the well known fact that there are only countably manyclones for \A\ — 2 [7]. In fact we do more. First we determine all maximal partial clonescontaining all permutations, and for three of them, we find a family of 2 ° subdonescontaining all the permutations. In two cases, such a family is contained in exactly onemaximal partial clone. These results show the substantial difference between the latticeof clones and the lattice of partial clones on a finite set.

A Survey on Intersections of Maximal Partial Clones of Boolean Partial Functions

We survey known results and present some new ones about intersections of maximal partial clones on a 2-element set.

Partial clones and their generating sets

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.

Caps and Colouring Steiner Triple Systems

Hill [6] showed that the largest cap in PG(5,3) has cardinality 56. Using this cap it is easy to construct a cap of cardinality 45 in AG(5,3). Here we show that the size of a cap in AG(5,3) is bounded above by 48. We also give an example of three disjoint 45-caps in AG(5,3). Using these two results we are able to prove that the Steiner triple system AG(5,3) is 6-chromatic, and so we exhibit the first specific example of a 6-chromatic Steiner triple system.

Generating sets for clones and partial clones

We briefly survey some known results on generating sets for clones, and then present results for partial clones established in Borner and Haddad, (1997). Namely, we give a criterion for recognizing not finitely generated strong partial clones on a finite set A and apply this criterion to a family of maximal partial clones over A.

On partial clones containing all idempotent partial operations

Let k>2 and k be a k-element set. We show that though the partial clone I:=/spl cap//sub a/spl isin/k/pPol {a} of all idempotent partial operations on k is contained in finitely many partial clones on k, it is quite hard to determine the number of all partial clones on k containing the partial clone I on k. Indeed this question is related to a hard problem in the area of extremal set theory.

Relation Graphs and Partial Clones on a 2-Element Set

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.

Characterization of Partial Sheffer Functions in 3-Valued Logic

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.