The Lattice of Closed Congruences on a Topological Lattice

Dennis J. Clinkenbeard
1981 Transactions of the American Mathematical Society  
Our primary objectives are: (1) if L is a lattice endowed with a topology making both the meet and join continuous then (i) the natural map which associates a congruence with the smallest topologically closed congruence containing it preserves finite meets and arbitrary joins; (ii) the lattice of such closed congruences is a complete Brouwerian lattice; (2) if L is a topological (semi) lattice with the unit interval as a (semi) lattice homomorphic image then the lattice of closed (semi) lattice
more » ... congruences has no compatible Hausdorff topology. Introduction. Given a topological lattice L, we investigate 9*(L) the lattice of all congruences on L which are topologically closed in L X L. This collection of congruences is particularly interesting when L is compact Hausdorff since each member of 9*(L) will then preserve many algebraic and topological properties of
doi:10.2307/1998361 fatcat:egckcmdnk5dd5blrbmaumgjs6u