Principal Topology on a Rees Matrix Semigroup using Green's Left Quasiorder

Abstract

This paper introduces a principal topology on a Rees matrix semigroup using Green's left quasiorder. Since principal topologies are in one-one correspondance with quasiorder relations on a set, the relations are commonly used for constructing such topologies. The basis for the topology is the collection of minimal open neighbourhoods corresponding to each element in a given set. When semigroups are considered with Green's left quasiorder, minimal open neighbourhoods are the principal left ideals. Hence, the collection of principal left ideals will turn out to be a basis for the principal topology on a semigroup. As long as a Rees matrix semigroup is considered, it is observed that these ideals exhibit certain interesting properties. This paper analyses these ideals in the context of a Rees matrix semigroup. The properties thus observed actually determine the number of elements in the so formed principal topology. Further, the topology hence obtained is an example for a finite topology on an infinite set, provided the order of the Rees matrices is finite.