Maximal partial clones with no finite basis (Q1966166)

E. Post described all clones on a 2-element set \(A\) and showed that every of these clones is finitely generated. This result does not hold for \(|A|\geq 3\). However, R. Freivalds has constructed a partial clone (i.e., a superposition closed subset of partial functions on \(A\) containing all projections) on a 2-element set which is not finitely generated. A partial clone is called strong if it contains all subfunctions of its partial functions. The authors give a criterion for recognizing not finitely generated strong partial clones and show that among all \(|A|+ 1\) maximal partial clones of Slupiecki type on \(A\), only one is finitely generated.