On the lower part of the lattice of partial clones

Miguel Couceiro (1) , Lucien Haddad (2) , Karsten Schölzel (3)

1 ORPAILLEUR - Knowledge representation, reasonning
2 RMCC - Royal Military College of Canada
3 Mathematics Research Unit

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.

Domaines: Informatique [cs]; Mathématiques [math]

Miguel Couceiro, Lucien Haddad, Karsten Schölzel. On the lower part of the lattice of partial clones. Journal of Multiple-Valued Logic and Soft Computing, 2019, 33 (3), pp.177-196. ⟨hal-01826870v4⟩