Representations of regular*-semigroups and their application to partial symmetry

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The purpose of this research is to study the representations of regular *-semigroups and their application to partial symmetry. During the term of our project, the head investigator obtained the following. For the results of other investigators and their detail, see the references in the larar.1. By introducing the concept "biarrowed word tree", we obtained that the free generalized inverse *-semigroup is isomorphic to the set of all biarrowed word tree in respect to the new product.2. The category of locally inverse *-semigroups and prehomomorphisms [homomorphisms] is isomorphic to the category of locally inductive *-groupoids and ordered functors [inductive functors].3. By introducing the concept "PG^*-quintet", we constructed a PG^semigroup. And we showed that a generalized inverse *-semigroup S is E-unitary if and only if S is isomorphic to a PG^*-semigroup.4. Any generalized inverse *-semigroup can be embedded in a *-complete, infinitely distributive generalized inverse *-semigroup.5. We characterized a congruence on a generalized inverse *-semigroup by using the concept "*-congruence pair"