Subdirectly irreducible semilattices with an automorphism. (English) Zbl 0770.08004

Let SA be the variety of all algebras with a semilattice operation \(\land\) and two unary operations \(f\) and \(f^{-1}\) such that \(f\) is an automorphism and \(f^{-1}\) is the inverse automorphism of the underlying semilattice. As an example take the algebra \({\mathcal P}(Z)\) defined on the set of all subsets of the set of integers \(Z\), where \(A\land B=A\cap B\); \(f(A)=\{a+1| a\in A\}\), and \(f^{-1}(A)=\{a-1| a\in A\}\).
This paper proves that every subdirectly irreducible algebra in SA can be embedded into the algebra \({\mathcal P}(Z)\) (hence all are countable). The main result of the paper describes the subdirectly irreducible algebras as members of some very specific intervals in the subalgebra lattice of \({\mathcal P}(Z)\). This deep result is too technical to be described here in detail.


08B26 Subdirect products and subdirect irreducibility
06A12 Semilattices
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