Connections between binary systems and admissible topologies

Let G = (a,b,c,...) be a groupoid and T a topology for G with U_a 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_c there exist U_a and U_b such that U_a·U_b c U_c. 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_c in T such that a·U_b ⊈ U_c for all U_b 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_c in T such that a·U_b ⊈ U_c and U_a·b ⊈ U_c for all U_a and U_b 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.

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Polytechnic Institute

Year dc:date.issued

1965

Author dc:creator

• Hanson, John Robert

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en

Handle dc:identifier.uri

http://hdl.handle.net/10919/104348

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/104348