Order-Theoretic and Ring-Theoretic Approaches to Inverse Semigroups.

Vagner suggested a problem in 1953: describe those (partially) ordered sets which support inverse semigroups. He also found a necessary condition for an ordered set to support an inverse semigroup. In the present work, it is proved that Vagner's condition is not sufficient. A minimal counterexample is found. All minimal examples of ordered set (S; <) with one of the following properties are also found: (1) S supports an inverse semigroup but does not support any Clifford semigroup. (2) S supports a Clifford semigroup but does not support any commutative inverse semigroup. The first example of ordered sets with property 1 was found by Dihtjar' in 1974, but it was not minimal. The order structure theorem and the ideal extension theorem for finite monogenic inverse semigroups are given in Chapter 2. Various authors considered this problem in ring theory: when is a regular ring strongly regular (=a regular ring with an inverse multiplicative semigroup)? Several new necessary and sufficient conditions are found. Moreover, weakly regular rings (=regular rings which are not strongly regular) are characterized: A regular ring R is weakly regular if and only if R contains a subring which is isomorphic to a 2 x 2 matrix ring over Z(,p) or Z. A weak identity is an identity which is satisfied by idempotents of a weakly regular ring. A characterization theorem for weak identities is proved: An identity is weak if and only if it is satisfied by idempotents of (Z(,2))(,2). Some classes of identities which are more general than the class of all weak identities are also characterized. To classify any given identity, a direct method is developed. It has been used successfully in a computer program which tests any identity.