Algebras of monads over the category of sets and over the catagory of topological spaces

Abstract

Algebrasof monads over ENS the category of sets and over TOP the category of topologicalspaces are studied in this thesis. There are sixteen induced monads Ft over ENS depending on t, obtained from sixteen adjunctions Ptop -j Gt : ORD7P --t ENS, where t is a list of some or all of four operations on ordered sets, denoted by 1\,V, T, 1... Categories of Ft-algebras ENSfl\vT 1., ENSfl\T.L, ENSfl\T and ENSfT.L are the categories of compact Hausdorff spaces, continuous semilattices, continuous lattices and completely distributive lattices respectively. All other categories of Fralgebras are variants of these. Fourmonads over TOP, the prime closed filter monad Wp, the prime open filter monad 1ip,the proper closed filter monad Wo and the proper open filter monad 1io, where Wp and 1ipare obtained by preimage adjunctions between LAT the category of lattices and TOP consideringclosed and open sets respectively and Wo and 1io are obtained by preimage adjun~tionsbetween MSLo the category of ORDATand TOP considering closed and open sets respectively are also studied. Categories of Wp-algebras and 1ip-algebras are compact ordered spaces, The categories of Wo-algebras and 1io-algebras are isomorphic with the f" " category of algebras ENS I\T of the proper filter monad FAT over ENS,