Chapter 1. Extension of the fundamental theorem of finite semigroups

This paper proves that some useful commutivity relations exist among semigroup wreath product factors that are either groups or combinatorial “units” U1, U2, or U3. Using these results it then obtains some characterizations of each of the classes of semigroups buildable from U1's, U2's, and groups (“buildable” meaning “dividing a wreath product of”). We show that up to division U1's can be moved to the right and U2's, and groups to the left over other units and groups, if it is allowed that the factors involved be replaced by their direct products, or in the case of U2, even by a wreath product. From this it is deduced that U1's and U2's do not affect group complexity, that any semigroup buildable from U1's, U2's, and groups has group complexity 0 or 1, and that all such semigroups can be represented, up to division, in a canonical form—namely, as a wreath product with all U1's on the right, all U2's on the left, and a group in the middle. This last fact is handy for developing characterizations. An embedding theorem for semigroups with a unique 0-minimal ideal is introduced, and from this and the commutivity results and some constructions proved for RLM semigroups, there is obtained an algebraic characterization for each class of semigroups that is a wreath product-division closure of some combination of U1's, U2's, and the groups. In addition it is shown, for i = 1,2,3, that if the unit Ui does not divide a semigroup S, then S can be built using only groups and units not containing Ui. Thus, it can be deduced that any semigroup which does not contain U3 must have group complexity either 0 or 1. This then establishes that indeed U3 is the determinant of group complexity, since it is already proved that both U1 and U2 are transparent with regard to the group complexity function, and it is known that with U3 (and groups) one can build semigroups with complexities arbitrarily large. Another conclusion is a combinatorial counterpart for the Krohn-Rhodes prime decomposition theorem, saying that any semigroups can be built from the set of units which divide it together with the set of those semigroups not having unit divisors. Further, one can now characterize those semigroups which commute over groups, showing a semigroup commutes to the left over groups iff it is “R1” (i.e., does not contain U1, i.e., is buildable form U2's and groups), and commutes to the right over groups iff it does not contain U2 (i.e., is buildable from groups and U1's). Finally, from the characterizations and their proofs one sees some ways in which groups can do the work of combinatorials in building combinatorial semigroups.