EMBEDDING THEOREMS FOR SEMIGROUPS WITH INVOLUTION

In Chapter 0 we start with the definitions of proper involutions on semigroups and rings, free products of semigroups with proper involution and we list some of their properties (most of them are known). In Chapter 1 we give a necessary and sufficient condition for a given semigroup with a proper involution to be embeddable into another which is also regular. In Chapter 2 we define amalgams of families of semigroups with involution, and give a necessary and sufficient condition for such an amalgam to be embeddable into a semigroup with a proper involution. In Chapter 3 we show that, given any inverse semigroup S, and any formally complex ring R, the natural involution on the semigroup ring R{S} is proper. We show further that R{S} has a zero nil radical when R is formally complex or a zero characteristic field. In Chapter 4 we give an example of a commutative inverse semigroup with a proper involution which cannot be embedded (i.e. while preserving the involution) into any ring with proper involution and also establish a necessary and sufficient condition for a given semigroup with proper involution to be embeddable into some ring with a proper involution