Multi algebras & related structures (1979)
AuthorsNolan, Francis Mauriceshow all
This thesis investigates the properties of two algebraic structures - multialgebras and partially ordered universal algebras. Multialgebras generalize the concept of a universal algebra to multivalued operations. Unlike universal algebras however there are many types of homomorphism associated with a multialgebra. A full homomorphism is defined and compared with the usual strong homomorphism of multialgebras. While they are similar in structure, the category of multialgebras and full homomorphisms is the dual of a category of boolean ordered algebras. This yields a more natural theory than that of the analogous category equivalent to multialgebras and strong homomorphisms. Partially ordered algebras are universal algebras with a partially ordered base. They are treated from the view point of a universal algebra with an additional unary multioperation (the partial ordering). In this fashion their peculiar properties can be explained by referring to either the universal algebra part or the multialgebra part. In particular the full and strong concepts of multialgebras are defined for the simpler structures and turn out to be dual notions.