Commutative unary algebras with join-semidistributive topology lattices (Q2207021)

A unary algebra \(\mathcal A=(A,\Omega)\) is called commutative if any two basic operations commute. For an algebra \(\mathcal A\), let \(\mathcal T(\mathcal A))\) be the set of all topologies on \(\mathcal A\). It is known that \(\mathcal T(\mathcal A))\) forms a complete lattice under inclusion. The main result of the paper is Theorem~1. Let \(\mathcal A=(A,\Omega)\) be a commutative unary algebra such that the lattice \(\mathcal T(\mathcal A))\) is join-semidistributive. Then either \(|A|=2\) or any two subalgebras generated by one element are comparable under inclusion. For the case when \(\mathcal A=(A,f)\) is a monounary algebra, a necessary and sufficient condition is proved under which the lattice \(\mathcal T(\mathcal A))\) is join-semidistributive: either \(\mathcal A\) consists of two 1-element cycles, or \(\mathcal A\) is generated by an element \(a\in A\) such that \(f^{n}(a)=f^{n+h}(a)\) for some \(n, h\in\mathbb N\), \(n\le 3\) (i.e., \(\mathcal A\) is an \(h\)-element cycle with a tail with at most three elements).