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 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
Journal of Automata, Languages and Combinatorics, Jul 1, 2001
The following two problems are addressed in this paper. Let $k \geq 2$, $\k$ be a $k$-element set... more The following two problems are addressed in this paper. Let $k \geq 2$, $\k$ be a $k$-element set and $M$ be a family of maximal partial clones with trivial intersection over $k$. 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 $k$, then what is the smallest possible cardinality of $F$? We show that the answer to these two problems is three.
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.
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.
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 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 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
Journal of Automata, Languages and Combinatorics, Jul 1, 2001
The following two problems are addressed in this paper. Let $k \geq 2$, $\k$ be a $k$-element set... more The following two problems are addressed in this paper. Let $k \geq 2$, $\k$ be a $k$-element set and $M$ be a family of maximal partial clones with trivial intersection over $k$. 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 $k$, then what is the smallest possible cardinality of $F$? We show that the answer to these two problems is three.
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.
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.
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.
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Papers by Lucien Haddad