Topology lattices of unary algebras (Q2759262)

Let \(A=(A,F)\) be an algebra. A topology on the set \(A\) in which all the operations in \(F\) are continuous is called a topology on algebra \(A\). It is obvious that the topologies on algebra \(A\) form a complete lattice, which is denoted by \(T(A)\). In this paper the following main results are presented.NEWLINENEWLINENEWLINE1) The dual lattice of the congruence lattice of an algebra can be embedded in the topology lattice of this algebra.NEWLINENEWLINENEWLINE2) Let \(A\) be a unary algebra and let \(B\) be a subalgebra of \(A\). Then \(T(B)\) can be embedded as a principal filter in the lattice \(T(A)\).NEWLINENEWLINENEWLINE3) Let \(A\) be a unar, i.e. an algebra with exactly one unary operation. Then the lattice \(T(A)\) is isomorphic to \(\text{Con}(A)\) iff \(A\) is a cycle.NEWLINENEWLINENEWLINE4) Unars \(A\) are characterized for which \(T(A)\) is a modular lattice, a chain, a lattice with complements or with pseudocomplements, respectively.