Extendable local partial clones
Let \(E\) be an infinite set, \(O(E)\), \(P(E)\) be the sets of all everywhere defined (respectively partial) operations on the set \(E\), \([F]\) be the clone (partial clone) generated by a set of operations \(F\subseteq O(E)\) (\(F\subseteq P(E)\)), Loc\((F)\) be the local closure of the set \(F\), defined by \(g\in \text{Loc}(F)\) iff for every finite non-single subset \(A\subset E\) such that the restriction \(g|_{A}\) is non-void there exists \(f\in[F]\) such that \(g|_{A}=f\) on Dom\((g|_{A})\). A (partial) clone \(\mathbf{A}\) is local if Loc\((\mathbf{A})=\mathbf{A}\). It is shown that a local partial clone \(\mathbf{B}\) is maximal in \(P(E)\) iff for every finite non-single subset \(A\subset E\) the restriction \(\mathbf{B}|_{A}\) either is the full partial clone \(P(A)\) or (at least for some \(A\)) is a maximal partial clone on \(A\). Let Pol\((\mathfrak{R}\)) denote the set of all partial operations which preserve a set of relations \(\mathfrak{R}\) defined on \(E\). A partial operation \(f\in\text{Pol}(\mathfrak{R})\) with finite domain is called extendable in a local partial clone \(\mathbf{B}=\text{Pol}(\mathfrak{R})\) if there exists \(g\in\mathbf{B}\cap O(E)\) such that \(f=g\) on \(\text{Dom}(f)\); a local partial clone is extendable if every partial operation from it with finite domain is extendable. It is shown that for a single relation \(R\) on the countable set \(E\) the partial clone \(\mathbf{B}=\text{Pol}(\mathfrak{R})\) is extendable iff each relation constructed by a \((\exists,\&,=)\)-formula from \(R\) is equivalent to a relation constructed by some \((\&,=)\)-formula. All partial orders \(x\leq y\) on the countable set such that the partial clone \(\text{Pol}(x\leq y)\) is extendable are characterized and a description of all extendable maximal local partial clones on the countable set is obtained.
scientific article; zbMATH DE number 5300633
scientific article