Steve Seif

COMBINATORIAL TECHNIQUES FOR DETERMINING RANK AND IDEMPOTENT RANK OF CERTAIN FINITE SEMIGROUPS

Let τ be a partition of the positive integer n. A partition of the set {1, 2 ,...,n } is said to be of type τ if the sizes of its classes form the partition τ of n. It is known that the semigroup S(τ ), generated by all the transformations with kernels of type τ , is idempotent generated. When τ has a unique non-singleton class of size d, the difficult Middle Levels Conjecture of combinatorics obstructs the application of known techniques for determining the rank and idempotent rank of S(τ ). We further develop existing techniques, associating with a subset U of the set of all idempotents of S(τ ) with kernels of type τ a directed graph D(U ); the directed graph D(U ) is strongly connected if and only if U is a generating set for S(τ ), a result which leads to a proof if the fact that the rank and the idempotent rank of S(τ ) are both equal to max n , n d +1 .

Algebra complexity problems involving graph homomorphism, semigroups and the constraint satisfaction problem

In this paper, we connect the constraint satisfaction problem with other complexity problems, like the polynomial equivalence problem for combinatorial 0-simple semigroups, the graph retraction problem and the geometry problem. We show that every constraint satisfaction problem is polynomially equivalent to an easily formulated algebra complexity problem. As an application we prove that the polynomial equivalence problem (word problem) for the 2 × 2 matrices over the two element field is co-NP-complete.

Counting techniques to label constant weight Gray codes with links to minimal generating sets of semigroups

Let τ be a partition of the positive integer n. A partition of the set Xn={1,2,…,n} is said to be of type τ if the sizes of its classes form the partition τ of n. Given 1<r<n, an r element subset A of Xn and a partition π of Xn are said to be orthogonal if every class of π meets A in exactly one element. Let Gn,r be the graph whose vertices are the r-element subsets of Xn, with two sets being adjacent if they intersect in r−1 elements. The graph Gn,r is Hamiltonian; Hamiltonian cycles of Gn,r are early examples of error-correcting codes, where they came to be known as constant weight Gray codes. A Hamiltonian cycle A1,A2,…,Anr in Gn,r is said to be orthogonally τ-labeled if there exists a list of distinct partitions π1,π2,…,πnr of type τ such that πi is orthogonal to both Ai and Ai+1 where i=1,2,…,nr taken modulo nr. For all but a finite class of partition types τ we present counting arguments to prove that any Hamiltonian cycle in Gn,r can be orthogonally τ-labeled. The remaining cases, with the exception of cases which are equivalent to the celebrated Middle Levels Conjecture, are treated via constructive arguments in a sequel. A semigroup of transformations of Xn is Sn-normal if it is closed under conjugation by the permutations of Xn. The combinatorics results here and in the sequel lead to a determination of the rank and idempotent rank of all Sn-normal semigroups, thereby broadly generalizing the known result that the rank and idempotent rank of K(n,r) is S(n,r), a Stirling number of the second kind.

Finite Normal Semigroups

n -element set Xn has only inner automorphisms is investigated. For a semigroup S of transformations of X n, let Gs denote the group of all permutations of Xn that preserve S under conjugation. Fix a group G of permutations of Xn and let T be a semigroup of transformations of Xn maximal with respect to the property GT = G. For certain groups G we give a complete description of all such T and prove that Aut T + Inn T ≅ G.

The poset on connected induced subgraphs of a graph need not be Sperner

SupposeG is a finite connected graph. LetC(G) denote the inclusion ordering on the connected vertex-induced subgraphs ofG. Penrice asked whetherC(G) is Sperner for general graphsG. Answering Penrices question in the negative, we present a treeT such thatC(T) is not Sperner. We also construct a related distributive lattice that is not Sperner.

Constructive techniques for labeling constant weight Gray codes with applications to minimal generating sets of semigroups

Abstract The Partition Type Conjecture, a generalization of the Middle Levels Conjecture of combinatorics, states that for all positive integers n and r , with n > r >1, and every non-exceptional partition type τ of weight r of X n , there exists a constant weight Gray code which admits an orthogonal labeling by partitions of type τ . We prove results in combinatorics and finite semigroup theory, providing the completion of the proof that the Partition Type Conjecture is true for all types having more than one class of size greater than one (and leaving open only those cases which are equivalent to the Middle Levels Conjecture). The rank of a finite semigroup S is the cardinality of a minimum generating set for S ; if S is idempotent generated, the idempotent rank of S is the cardinality of a minimum idempotent generating set for S . A semigroup of transformations of X n ={1,…, n } is said to be S n -normal if S is closed under conjugation by the permutations of X n . The results here concerning the Partition Type Conjecture are used to determine a simple formula for the rank and idempotent rank of every S n -normal semigroup.

ALGORITHMS FOR LABELING CONSTANT WEIGHT GRAY CODES

Let n and r be positive integers with 1 < r < n, and let Xn = {1, 2 ,..., n} .A nr-set A and a partition π of Xn are said to be orthogonal if every class of π meets A in exactly one element. We prove that if A1, A2 ,..., A ( n) is a list of the distinct r-sets of Xn with |Ai ∩ Ai+1 |= r − 1f ori = 1, 2 ,..., n r taken modulo n r , then there exists a list of distinct partitions π1 ,π 2 ,...,π ( n) such that πi is orthogonal to both Ai and Ai+1. This result states that any constant weight Gray code admits a labeling by distinct orthogonal partitions. Using an algorithm from the literature on Gray codes, we provide a surprisingly efficient algorithm that on input (n, r) outputs an orthogonally labeled constant weight Gray code. We also prove a two-fold Gray enumeration result, presenting an orthogonally labeled constant weight Gray code in which the partition labels form a cycle in the covering graph of the lattice of all partitions of Xn. This leads to a conjecture related to the Middle Levels Conjecture. Finally, we provide an application of our results to calculating minimal generating sets of idempotents for finite semigroups.

Permutational labelling of constant weight Gray codes

We prove that for positive integers n and r satisfying 1 < r < n, with the single exception of n = 4 and r = 2, there exists a constant weight Gray code of r-sets of Xn = {1,2,... ,n} that admits an orthogonal labelling by distinct partitions, with each subsequent partition obtained from the previous one by an application of a