On clones of operations over finite Boolean algebras
Let \(\mathcal{O}_{A}=\bigcup_{n\geq 1}A^{A^{n}}\) denote the set of all finitary operations on \(A\). A set \(\mathcal{C}\subseteq \mathcal{O}_{A}\) of finitary operations is a clone of operations on \(A\) if it contains all the projection maps \(\pi_{i}^{n}:A^{n}\to A,\langle x_{1},\dots,x_{n}\rangle\mapsto x_{i}\), and is closed with respect to superposition of operations in the following sense: for all \(f,g_{1},\dots,g_{n}\in \mathcal{C}\) such that \(\text{ar}(f)=n\) and \(\text{ar}(g_{1})=\dots=\text{ar}(g_{n})\) we have \(f(g_{1},\dots,g_{n})\in \mathcal{C}\). For every set \(F\subseteq \mathcal{O}_{A}\) there is a least clone containing \(F\). It is referred to as the clone generated by \(F\) and denoted by Clo(\(F\)). For a finite Boolean algebra \(\mathbf{B}\) let \(\mathcal{O}_{\mathbf{B}}=\bigcup_{n\geq 1}\text{hom}(\mathbf{B}^{n},\mathbf{B})\) denote the set of all finitary operations over \(\mathbf{B}\). The set \(\mathcal{O}_{\mathbf{B}}\) is a clone of operations in a usual sense. A clone \(\mathcal{C}\subseteq \mathcal{O}_{\mathbf{B}}\) will be referred to as a clone of operations over \(\mathbf{B}\). Let \(\mathcal{L}_{\mathbf{B}}\) denote the lattice of clones of operations over \(\mathbf{B}\). A dual operation over a finite set \(X\) is any mapping \(f:X\to m\cdot X\), where \(m\cdot X=\{1,\dots,m\}\times X\) is the \(m\)th copower of \(X\). We say that \(m\) is the \textit{arity} of \(f\) and write \(\text{ar}(f)=m\). Every dual operation \(f:X\to m\cdot X\) gives rise to two maps, val\(f:X\to X\), the value of \(f\), and lbl\(f:X\to \{1,\dots,m\}\), the labelling of \(f\), defined by \(f(x)=\langle j,y\rangle\) if and only if val\(f(x)=y\) and lbl\(f(x)=j\). Let \(\mathcal{\overline{O}}_{X}=\bigcup_{n\geq 1}(n\cdot X)^{X}\) denote the set of all finitary dual operations over \(X\), and \(\mathcal{\overline{J}}_{X}\) the set of all injection maps \(\iota_{i}^{n}:X\to n\cdot X, x\mapsto \langle i,x\rangle, n\in \mathbf{N}, 1\leq i\leq n\). A set \(\mathcal{C}\subseteq \mathcal{\overline{O}}_{X}\) of finitary dual operations is a clone of dual operations over \(X\) if \(\mathcal{\overline{J}}_{X}\subseteq \mathcal{C}\) and \(\mathcal{C}\) is closed with respect to superposition of dual operations in the following sense: for all \(f,g_{1},\dots,g_{n}\in \mathcal{C}\) such that \(\text{ar}(f)=n\) and \(\text{ar}(g_{1})=\dots=\text{ar}(g_{n})=k\) we have \([g_{1},\dots,g_{n}]\circ f\in \mathcal{C}\) (where \([g_{1},\dots,g_{n}] : n\cdot X\to k\cdot X\), \([g_{1},\dots,g_{n}](\langle j,x\rangle)=g_{j}(x)\). By \(\mathcal{\overline{L}}_{X}\) we denote the lattice of clones of dual operations over \(X\). If \(\mathbf{B}\) is a finite Boolean algebra and \(X\) the set of its atoms, then \(\mathcal{L}_{\mathbf{B}}\cong \mathcal{\overline{L}}_{X}\). Maximal clones in \(\mathcal{L}_{\mathbf{B}}\), therefore, correspond to maximal clones of dual operations over \(X\). The author shows that the lattice of clones of operations over a finite Boolean algebra (\(\mathcal{L}_{\mathbf{B}}\)) is complemented (in contrast to the lattice of clones of operations over a finite set) by showing that \(\mathcal{\overline{L}}_{X}\) is a complemented lattice (Theorem 3.7). Finally, the author introduces the notion of iterative algebra of operations (\(\mathbf{A}_{\mathbf{B}}\)) and shows that it admits only inner automorphisms (Theorem 4.3) and has precisely three congruences (Theorem 4.8) (very much like the iterative algebra of operations over a finite set).
scientific article; zbMATH DE number 5508759