Dichotomy on intervals of strong partial Boolean clones

Abstract

The following result has been shown recently in the form of a dichotomy: For every total clone $C$ on $\mathbf{2} := \{0,1\}$, the set $\mathcal{I}(C)$ of all partial clones on $\mathbf{2}$ whose total component is $C$, is either finite or of continuum cardinality. In this paper we show that the dichotomy holds, even if only strong partial clones are considered, i.e., partial clones which are closed under taking subfunctions: For every total clone $C$ on $\mathbf{2}$, the set $\mathcal{I}_{\mathrm{Str}}(C)$ of all strong partial clones on $\mathbf{2}$ whose total component is $C$, is either finite or of continuum cardinality.