Pierre Antoine Grillet

The structure of regular semigroups, III: The reduced case

Reduced regular semigroups were characterized in the previous announcements of this sequence: up to isomorphism, they are the full regular subsemigroups of certain semigroups U built from regular partially ordered sets. We now complete this result by an explicit procedure to pick out all full regular subsemigroups of any U. This also provides a way to obtain (from partially ordered sets) all possible egg-box pictures of regular semigroups, as well as a blueprint for the multiplication in the general case.

Computing finite commutative semigroups

Precedence results are used to improve existing algorithms for the enumeration of finite commutative semigroups. As an application 11,545,843 distinct commutative semigroups of order 9 were found.

Commutative semigroup cohomology

Triple cohomology for commutative semigroups is described in concrete terms and related to commutative group coextensions.

Completely 0-simple semirings

A completely (0-) simple semiring is a semiring R which is (0-) simple and is the union of its (0-) minimal left ideals and the union of its (0-) minimal right ideals. Structure results are obtained for such semirings. First the multiplicative semigroup of R is completely (0-) simple; for any Jf-class 7/(^0), H (u {0}) is a subsemiring. If furthermore R has a zero but is not a division ring, and if (H <J {0}, +) has a completely simple kernel for some H as above (for instance, if R is compact or if the ^-classes are finite), then (i) {R, + ) is idempotent ; (ii) R has no zero divisors, additively or multiplicatively. Additional results are given, concerning the additive ./-classes of R and also (0-) minimal ideals of semirings in general. A (0-) simple semiring R which has (0-) minimal left ideals and (0-) minimal right ideals need not be the set-theoretical union of its (0-) minimal left ideals, nor the union of its (0-) minimal right ideals; when R is equal to both unions, we say that it is completely (0-) simple. Our purpose is to describe the structure of these semirings. We prove that the multiplicative semigroup (/?, ■) of a completely 0-simple semiring R is a completely 0-simple semigroup. In case the ^-classes of R (which in this case coincide with that of (/?, •)) ar^ finite, it turns out that, if R is not a division ring, then (/?, +) is an idempotent semigroup and furthermore R has no zero divisors. In this case, we also give an explicit description of the addition of R in each ^-class of (/?, +) and study how it relates to the multiplication and the addition in other ^-classes. A complete explicit description of the addition of R is not attempted, since any idempotent semigroup is the additive semigroup of some completely simple semiring. The first section contains a number of basic results on 0-minimal ideals in semirings, culminating with the result that, if a 0-minimal two-sided ideal M of a semiring R with zero contains a 0-minimal left ideal of R, then it is generated by the union of its own 0-minimal left ideals (which coincide with the 0-minimal left ideals of R contained in M) (Theorem 1.9). Some of these results are used in §2 to give various characterizations of completely 0-simple semirings; namely, the following conditions are equivalent for any semiring R : Received by the editors December 22, 1969. AMS 1969 subject classifications. Primary 1696.

The commutative cohomology of finite semigroups

Abstract The second cohomology group of any commutative semigroup is computed from its presentation; special attention is paid to the finite case.

Cocycles in commutative semigroup cohomology

An alternate description of triple cohomology for commutative semigroups is given in dimensions 1,2, and 3

The commutative cohomology of nilsemigroups

Abstract Our main result computes the second cohomology group of any finite commutative nilsemigroup with identity adjoined, in terms of its presentation.

Nilsemigroups on trees

We show how to reconstruct finite commutative nilsemigroups from their underlying partially ordered sets, and characterize which finite trees underlie finite commutative nilsemigroups.

Computing finite commutative semigroups: Part III

New precedence results are obtained for semigroups with zero, which describe the first non-zero row in detail and provide effective comparisons with subsequent rows. These results are used to further sharpen existing algorithms for the computation of finite commutative semigroups, confirm the number of distinct commutative semigroups of order 9, and determine the number of distinct commutative semigroups of order 10.

Isomorphisms of one-relator semigroup algebras

All isomorphisms between comutative semigroup algebras K[S] and K[T] are found when K is a field and S, T have two generators subject to a single homogeneous defining relation