Theses and Dissertations (Mathematics)http://hdl.handle.net/10386/672017-11-18T13:41:03Z2017-11-18T13:41:03ZOn yosida frames and related framesMatabane, Mogalatjane Edwardhttp://hdl.handle.net/10386/7662017-11-16T10:03:12Z2012-01-01T00:00:00ZOn yosida frames and related frames
Matabane, Mogalatjane Edward
Topological structures called Yosida frames and related algebraic frames are studied in the realm of Pointfree Topology. It is shown that in algebraic frames regular elements are those for which compact elements are rather below the regular elements, and algebraic frames are regular if and only if every compact element is rather below itself if and only if the frame has the Finite Intersection Property (FIP) and each prime element is minimal.
We also show that Yosida frames are those algebraic frames with the Finite Intersection Property and are finitely subfit; that these frames are also those semi-simple algebraic frames with FIP and a disjointification where dim (L)≤ 1; and we prove that in an algebraic frame with FIP, it holds that dom (L) = dim (L). In relation to normality in Yosida frames, we show that in a coherent normal Yosida frame L, the frame is subfit if and only if it is regular if and only if it is zero- dimensional if and only if every compact element is complemented.
Thesis (MA. (Mathematics)) -- University of Limpopo, 2012
2012-01-01T00:00:00ZOn closed and quotient maps of localesThoka, Mahuleng Ludwickhttp://hdl.handle.net/10386/772017-11-17T01:00:23Z2007-01-01T00:00:00ZOn closed and quotient maps of locales
Thoka, Mahuleng Ludwick
The category Loc of locales and continuous maps is dual to the category
Frm of frames and frame homomorphisms. Regular subobjects of a locale A
are elements of the form
Aj = fj : A ! A j j(a) = ag:
The subobjects of this form are called sublocales of A. They arise from the
lattice OX of open sets of a topological space X in a natural way. The right
adjoint of a frame homomorphism maps closed (dually, open) sublocales to
closed (dually, open) sublocales.
Simple coverings and separated frames are studied and conditions under
which they are closed (or open) are those that are related to coequalizers
are shown. Under suitable conditions, simple coverings are regular epimorphisms.
Extremal epimorphisms and strong epimorphisms in the setting of locales are
studied and it is shown that strong epimorphisms compose. In the category
Loc of locales and continuous maps, closed surjections are regular epimorphisms
at least for those surjections with subfit domains.
Thesis (M.Sc. (Mathematics)) --University of Limpopo, 2007
2007-01-01T00:00:00Z