Decidability of Krohn-Rhodes complexity c=1c = 1 of finite semigroups and automata

When decomposing a finite semigroup into a wreath product of groups and aperiodic semigroups, complexity measures the minimal number of groups that are needed. Determining an algorithm to compute complexity has been an open problem for almost 60 years. The main result of this paper proves decidability of Krohn-Rhodes complexity $c = 1$ of finite semigroups and automata. This is achieved by showing the lower bounds in work by Henckell, Rhodes and Steinberg from 2012 is sharp using profinite methods and results of McCammond from 1991 and 2001.Comment: 34 pages; substantial additions, Sections 3-5 are new compared to version tw

The rank of the semigroup of transformations stabilising a partition of a finite set

Let $\mathcal{P}$ be a partition of a finite set $X$. We say that a full transformation $f:X\to X$ preserves (or stabilizes) the partition $\mathcal{P}$ if for all $P\in \mathcal{P}$ there exists $Q\in \mathcal{P}$ such that $Pf\subseteq Q$. Let $T(X,\mathcal{P})$ denote the semigroup of all full transformations of $X$ that preserve the partition $\mathcal{P}$. In 2005 Huisheng found an upper bound for the minimum size of the generating sets of $T(X,\mathcal{P})$, when $\mathcal{P}$ is a partition in which all of its parts have the same size. In addition, Huisheng conjectured that his bound was exact. In 2009 the first and last authors used representation theory to completely solve Hisheng's conjecture. The goal of this paper is to solve the much more complex problem of finding the minimum size of the generating sets of $T(X,\mathcal{P})$, when $\mathcal{P}$ is an arbitrary partition. Again we use representation theory to find the minimum number of elements needed to generate the wreath product of finitely many symmetric groups, and then use this result to solve the problem. The paper ends with a number of problems for experts in group and semigroup theories

Schur-Weyl dualities for symmetric inverse semigroups

We obtain Schur-Weyl dualities in which the algebras, acting on both sides, are semigroup algebras of various symmetric inverse semigroups and their deformations.Comment: 14 page

Chapter 1. Extension of the fundamental theorem of finite semigroups

AbstractThis 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 charactérizations.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

Orbits of primitive k-homogenous groups on (N − k)-partitions with applications to semigroups

© 2018 American Mathematical Society. The purpose of this paper is to advance our knowledge of two of the most classic and popular topics in transformation semigroups: automorphisms and the size of minimal generating sets. In order to do this, we examine the k-homogeneous permutation groups (those which act transitively on the subsets of size k of their domain X) where |X| = n and k < n/2. In the process we obtain, for k-homogeneous groups, results on the minimum numbers of generators, the numbers of orbits on k-partitions, and their normalizers in the symmetric group. As a sample result, we show that every finite 2-homogeneous group is 2-generated. Underlying our investigations on automorphisms of transformation semigroups is the following conjecture: If a transformation semigroup S contains singular maps and its group of units is a primitive group G of permutations, then its automorphisms are all induced (under conjugation) by the elements in the normalizer of G in the symmetric group. For the special case that S contains all constant maps, this conjecture was proved correct more than 40 years ago. In this paper, we prove that the conjecture also holds for the case of semigroups containing a map of rank 3 or less. The effort in establishing this result suggests that further improvements might be a great challenge. This problem and several additional ones on permutation groups, transformation semigroups, and computational algebra are proposed at the end of the paper