Let G = (a,b,c,...) be a groupoid and T a topology for G with U<sub>a</sub> denoting an open set in T that contains the element a. The topology T is admissible for G if for every a·b=c and U<sub>c</sub> there exist U<sub>a</sub> and U<sub>b</sub> such that U<sub>a</sub>·U<sub>b</sub> c U<sub>c</sub>. G is said to be topologically trivial if the only admissible topologies for G are the discrete and indiscrete. It is shown that finite groups are topologically trivial if and only if they are simple. It is shown that finite topologically trivial semigroups are necessarily groups. Various classes of topologically trivial groupoids are examine, and it is shown that there exist topologically trivial groupoids of every order.
G is said to be right (analogously left) topologically trivial if one can find elements a·b = c in G and U<sub>c</sub> in T such that a·U<sub>b</sub> ⊈ U<sub>c</sub> for all U<sub>b</sub> in T whenever T is not trivial. G is said to be totally topologically trivial if one can find a·b = c in G and U<sub>c</sub> in T such that a·U<sub>b</sub> ⊈ U<sub>c</sub> and U<sub>a</sub>·b ⊈ U<sub>c</sub> for all U<sub>a</sub> and U<sub>b</sub> in T whenever T is not trivial. Right, left, and total topologically triviality are studies for various algebraic systems.
A continuity condition that always holds is exhibited as are new proofs for several old theorems.
Consequences of imposing the tower topology on various algebraic systems are examined.
If the proper subset I contained in the groupoid G is such that the null set, the set G, and each singleton set of the elements in G-I form the basis for an admissible topology for G, then I is called a generalized ideal in G. Properties of generalized ideals are studied at length.
A function t from a groupoid G to another groupoid is called a local homomorphism if for each a and b in G there exist r and s in G such that a·b = r·s and such that t(r·s) = t(r)·t(s). Several properties of local homomorphisms are examined. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/104348 |
| Date | January 1965 |
| Creators | Hanson, John Robert |
| Contributors | Mathematics |
| Publisher | Virginia Polytechnic Institute |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | 84 leaves, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 20269181 |
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