Near unanimity: An obstacle to general duality theory
Let \(\underline P= (P; F)\) be an algebra. Let \(\underset\widetilde{} P= (P; G, H, R, T)\) be a structure such that all \(n\)-ary (partial) operations from \(G\) \((H)\) are homomorphisms of the \(n\)th power of \(\underline P\) (of domain) into \(\underline P\), all \(n\)-ary relations from \(R\) are subuniverses of \(\underline P\), and \(T\) is the discrete topology on \(P\). Let \({\mathcal A}= \text{ISP}(\underline P)\) and \({\mathcal X}= \text{IS}_{\text{c}} \text{P}(P)\) (\(\text{S}_{\text{c}}\) means closed substructures). There are well-defined hom-functors \(D: {\mathcal A}\to {\mathcal X}\) and \(E: {\mathcal X}\to {\mathcal A}\). If \(ED(\underline A)\) is an isomorphism for all \(\underline A\in {\mathcal A}\), we say that \(\underline P\) admits natural duality (\(\underset\widetilde{} P\) yields a natural duality) in the sense of B. A. Davey. It was proved by B. A. Davey and H. Werner that if \(\underline P\) has a term (near- unanimity term) of arity \(\geq 3\) such that \(T(x,\dots, x, y)= T(x,\dots, y, x)= T(x, y, \dots, x)= T(y, \dots, x)\) are identities in \(\underline P\), then \(\underline P\) admits a natural duality. The authors prove that in the finite case \(\underline P\) admits a natural duality and has a distributive lattice of congruences iff \(\underline P\) has a near-unanimity term iff \(\underline P\) has a distributive lattice of congruences and for every compact totally disconnected topological space \(X\) each subalgebra of the algebra \(C(X, \underline P)\) of all continuous functions is closed in the pointwise topology. Some other connections between these notions are given.
scientific article; zbMATH DE number 759392
scientific article