On the interval of strong partial clones of Boolean functions containing Pol({(0,0),(0,1),(1,0)})
Identifieur interne : 000403 ( Main/Exploration ); précédent : 000402; suivant : 000404On the interval of strong partial clones of Boolean functions containing Pol({(0,0),(0,1),(1,0)})
Auteurs : Miguel Couceiro [France] ; Lucien Haddad [Canada] ; Karsten Schölzel [Luxembourg (pays)] ; Tamas Waldhauser [Hongrie]Source :
Abstract
D. Lau raised the problem of determining the cardinality of the set of all partial clones of Boolean functions whose total part is a given Boolean clone. The key step in the solution of this problem, which was obtained recently by the authors, was to show that the sublattice of strong partial clones on {0, 1} that contain all total functions preserving the relation ρ 0,2 = {(0, 0), (0, 1), (1, 0)} is of continuum cardinality. In this paper we represent relations derived from ρ 0,2 in terms of graphs, and we define a suitable closure operator on graphs such that the lattice of closed sets of graphs is isomorphic to the dual of this uncountable sublattice of strong partial clones. With the help of this duality, we provide a rough description of the structure of this lattice, and we also obtain a new proof for its uncountability.
Url:
Affiliations:
- Canada, France, Hongrie, Luxembourg (pays)
- Grand Est, Lorraine (région)
- Metz, Nancy
- Université de Lorraine
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key="FR"></country></address></desc></org></tutelle></tutelles></hal:affiliation><country>Hongrie</country></affiliation></author></analytic></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">D. Lau raised the problem of determining the cardinality of the set of all partial clones of Boolean functions whose total part is a given Boolean clone. The key step in the solution of this problem, which was obtained recently by the authors, was to show that the sublattice of strong partial clones on {0, 1} that contain all total functions preserving the relation ρ 0,2 = {(0, 0), (0, 1), (1, 0)} is of continuum cardinality. In this paper we represent relations derived from ρ 0,2 in terms of graphs, and we define a suitable closure operator on graphs such that the lattice of closed sets of graphs is isomorphic to the dual of this uncountable sublattice of strong partial clones. With the help of this duality, we provide a rough description of the structure of this lattice, and we also obtain a new proof for its uncountability.</div></front></TEI><affiliations><list><country><li>Canada</li><li>France</li><li>Hongrie</li><li>Luxembourg (pays)</li></country><region><li>Grand Est</li><li>Lorraine (région)</li></region><settlement><li>Metz</li><li>Nancy</li></settlement><orgName><li>Université de Lorraine</li></orgName></list><tree><country name="France"><region name="Grand Est"><name sortKey="Couceiro, Miguel" sort="Couceiro, Miguel" uniqKey="Couceiro M" first="Miguel" last="Couceiro">Miguel Couceiro</name></region></country><country name="Canada"><noRegion><name sortKey="Haddad, Lucien" sort="Haddad, Lucien" uniqKey="Haddad L" first="Lucien" last="Haddad">Lucien Haddad</name></noRegion></country><country name="Luxembourg (pays)"><noRegion><name sortKey="Scholzel, Karsten" sort="Scholzel, Karsten" uniqKey="Scholzel K" first="Karsten" last="Schölzel">Karsten Schölzel</name></noRegion></country><country name="Hongrie"><noRegion><name sortKey="Waldhauser, Tamas" sort="Waldhauser, Tamas" uniqKey="Waldhauser T" first="Tamas" last="Waldhauser">Tamas Waldhauser</name></noRegion></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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