Small semilattices and costability



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This dissertation centers its attention firstly on topological semilattices with small semi lattices and some equivalences of that property. This property is highly related to finite dimensional locally connected compact topological semi lattices and the extension property with respect to finite subsemilattices. A topological semigroup S is said to be costable with respect to a class of semigroups if for each semigroup T in the class and given a continuous homomorphism from T onto S, then cd T > cd S where cd is the codimension function. The basic example of a compact semilattice which is costable with respect to compact semilattices is a 1-dimensional compact semilattice that fails to have small semilattices. With this basic example, we can construct higher dimensional costable semilattices. Although costability stems from looking at semi lattices, there is no reason why we cannot consider compact semigroups in general. Chapter 4 is intended to illustrate various classes of costable semigroups, and hopefully, these classes can characterize some of the commutative costable semigroups.