Inessa Levi

AUTOMORPHISMS OF NORMAL TRANSFORMATION SEMIGROUPS

Let X be an infinite set, be the group of all bijections of X and S be a semigroup of total transformations of X with the composition of transformations f and g in S defined by the formula We say that S is a - normal semigroup if

Normal semigroups of one-to-one transformations

If a transformation semigroup S is defined by means of certain properties there is aproblem of determining the elements of S explicitly. In this paper, the above problem isconsidered for ^-normal semigroups S of total one-to-one transformations of a set X.A transformation semigroup S on X is termed ^-norma

Automorphisms of normal partial transformation semigroups

We let X be an arbitrary infinite set. A semigroup S of total or partial transformations of X is called -normal if hSh -1 = S , for all h in , the symmetric group on X . For example, the full transformation semigroup , the semigroup of all partial transformations , the semigroup of all 1–1 partial transformations and all ideals of and are -normal.

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 .

A n -normal Semigroups

Abstract. No Abstract.

THE SEMIGROUP OF ONE-TO-ONE TRANSFORMATIONS WITH FINITE DEFECTS

Let if be the semigroup of all total one-to-one transformations of an infinite set X. For an / e V let the deject of/, def/, be the cardinality of X R(J), where R(f) =f(X) is the range of /. Then Sf is a disjoint union of the symmetric group ^x on X, the semigroup 5 of all transformations in y with finite non-zero defects and the semigroup S of all transformations in 5 with infinite defects, such that 5 U 5 and S are ideals of Sf. The properties of *3X a d S have been investigated by a number of authors (for the latter it was done via Baer-Levi semigroups, see [2], [3], [5], [6], [7], [8], [9], [10] and note that S decomposes into a union of Baer-Levi semigroups). Our aim here is to study the semigroup S. It is not difficult to see that S is left cancellative (we compose functions /, g in 5 as/g(x) =/(g(*)), for x e X) and idempotent-free. All automorphisms of 5 are inner [4], that is of the form f^hfh~\f€S,he <SX. In the present paper, we are concerned with congruences, Greens relations and ideals of 5. A large variety of distinct types of congruences on S is present and the main results are the content of Theorems 4, 5 and 6. In the concluding remark we state some unsolved problems and conjectures on congruences on 5. For/, g e S, let D(J, g) = {* :f(x)3=g(x)}. The next lemma is easily verified.

A mathematical model for interpersonal relationships in social networks

This paper uses the mathematical structure of algebraic semigroups to model interpersonal relationships. The basis for the model is the periodic surveys administered to each member of a social network. The model uses individual preference to uncover the underlying structure of the team. The model may possibly be used to study the stability of the network.

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.

Ranks of Semigroups Generated by Order-Preserving Transformations with a Fixed Partition Type

Abstract The rank of a finite semigroup is the minimal size of its generating set. Let T n be the semigroup of all total transformations of the set {1, 2,…, n}, and let 𝒪 n be the subsemigroup of T n of all the order-preserving transformations whose images consist of at most n − 1 elements. We study the subsemigroups S 𝒪(τ) of 𝒪 n generated by the order-preserving transformations whose kernels are partitions of X n of a given partition type τ. We characterize idempotent-generated semigroups S 𝒪(τ), and show that S 𝒪(τ) is idempotent-generated precisely when it is regular, and that occurs if and only if the weight r of τ is n − 1 or 1. For arbitrary partition type τ of weight r we determine the Greens relations on S 𝒪(τ), and we show that its rank equals to .

Group closures of one-to-one transformations

For a semigroup S of transformations of an infinite set X let G s be the group of all the permutations of X that preserve S under conjugation. Fix a permutation group H on X and a transformation f of X , and let 〈 f : H 〉 = 〈{ hfh −1 : h ∈ H }〉 be the H -closure of f . We find necessary and sufficient conditions on a one-to-one transformation f and a normal subgroup H of the symmetric group on X to satisfy G 〈 f : H 〉 = H . We also show that if S is a semigroup of one-to-one transformations of X and G S contains the alternating group on X then Aut( S ) = Inn( S ) ≅ G S .