On Idempotent Ranks of Semigroups of Partial Transformations

A subset U of a semigroup S is a generating set for S if every element of S may be written as a finite product of elements of U. The rank of S is the size of a minimal generating set of S, and the idempotent rank of S is the size of a minimal generating set of S consisting of idempotents in S. A partition of a q-element subset of the set X_n={1,2,..., n} is said to be of type if the sizes of its classes form the partition of q n. A non-trivial partition of a positive integer q consists of k < q elements. For a non-trivial partition of q n, the semigroup S(), generated by all the transformations with kernels of type , is idempotent-generated. It is known that if is a non-trivial partition of n, that is, S() consists of total many-to-one transformations, then the rank and the idempotent rank of S() are both equal to max{ⁿ_d, N()}, where N() is the number of partitions of X_n of type . We extend this result to semigroups of partial transformations, and prove that if is a non-trivial partition of q < n, then the rank and the idempotent rank of S() are both equal to N().     

Non-group Ranks in Finite Full Transformation Semigroups

T _n be the full transformation semigroup on a finite set. Both rank and idempotent rank of the semigroup K(n,r) = {α∈T _n : | im α | ≤r, 2 ≤ r ≤ n - 1. In this paper we prove that the non-group rank, defined as the cardinality of a minimal generating set of non-group elements, of K(n,r) is S(n,r) , the Stirling number of the second kind.     

The rank of the endomorphism monoid of a uniform partition

The rank of a semigroup is the cardinality of a smallest generating set. In this paper we compute the rank of the endomorphism monoid of a non-trivial uniform partition of a finite set, that is, the semigroup of those transformations of a finite set that leave a non-trivial uniform partition invariant. That involves proving that the rank of a wreath product of two symmetric groups is two and then use the fact that the endomorphism monoid of a partition is isomorphic to a wreath product of two full transformation semigroups. The calculation of the rank of these semigroups solves an open question.     

A note on the rank of semigroups

The rank of a semigroup $\mathcal{A}The rank of a semigroup A\mathcal{A} of functions from a finite set X to X is the minimum of |f(X)| over f ? Af\in \mathcal{A}. Given a finite set X and a subset Y of X, we show that if A\mathcal{A} is a semigroup of functions from X to X and ℬ a transitive semigroup of functions from Y to Y, then the rank of A\mathcal{A} divides that of ℬ provided that f(X)⊆Y for some f ? Af\in \mathcal{A} and that each function in ℬ is the restriction of a function in A\mathcal{A} to Y. To prove this, we generalize a result of Friedman which says that one can partition Y into q subsets of equal weight where q is the rank of ℬ. When one extends a transitive automaton by adding new states and letters, a similar condition guarantees that the rank of the extension divides the original rank.     

The Word Problem for Relatively Free Semigroups

An example of a series of varieties of semigroups X_p with the finite basis property is constructed for which the word problem in the relatively free semigroup F_n X_p of rank n in the variety X_p is decidable if and only if n < p.     

Some examples of semigroup algebras of finite representation type

The semigroup algebras over a field K of the semigroups T_n of all permutations of a set of n elements are considered. It is proved: if n≤3 and (n!)^-1∈ K then the algebra KT_n has a finite representation type. Also the finiteness of the representation type of the semigroup algebra KS is established, where S is the sub-semigroup of T_n (n is arbitrary) such that S=J_n∪G where J_n={x∈T_n|rank x=1}, while G is a doubly transitive subgroup of the symmetric group S_n, the order of G being invertible in K. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 229–238, 1987.     

Approximate Amenability of Certain Semigroup Algebras

In the present paper, it is shown that a left cancellative semigroup S (not necessarily with identity) is left amenable whenever the Banach algebra ℓ¹(S) is approximately amenable. It is also proved that if S is a Brandt semigroup over a group G with an index set I, then ℓ¹(S) is approximately amenable if and only if G is amenable. Moreover ℓ¹(S) is amenable if and only if G is amenable and I is finite. For a left cancellative foundation semigroup S with an identity such that for every M_a(S)-measurable subset B of S and s ∈ S the set sB is M_a(S)-measurable, it is proved that if the measure algebra M_a(S) is approximately amenable, then S is left amenable. Concrete examples are given to show that the converse is negative.     

Baer extensions of rings and stone extensions of semigroups

Let R be a commutative semigroup [resp. ring] with identity and zero, but without nilpotent elements. We say that R is a Stone semigroup [Baer ring], if for each annihilator ideal P⊂R there are idempotents e₁ ε P and e₂ ε Ann(P) such that x→(e₁x, e₂x):R→P×Ann(P) is an isomorphism. We show that for a given R there exists a Stone semigroup [Baer ring] S containing R that is minimal with respect to this property. In the ring case, S is uniquely determined if one requires that there be a natural bijection between the sets of annihilator ideals of R and S. This is close to results of J. Kist [5]. Like Kist, we use elementary sheaf-theoretical methods (see [2], [3], [6]). Proofs are not very detailed. An address delivered at the Symposium on Semigroups and the Multiplicative Structure of Rings, University of Puerto Rico, Mayaguez, Puerto Rico, March 9–13, 1970.     

A Generalization of Symmetric Inverse Semigroups

The set of difunctional binary relations D_X plays a special role in representing inverse semigroups by binary relations. However, D_X is not an inverse semigroup either with the standard operation ∘, or with an alternative operation introduced in [6]. We introduce a new binary operation ⋄ on the set B_X of binary relations. We demonstrate that (D_X, ⋄) is an inverse semigroup, and the symmetric inverse semigroup (I_X, ∘) is a subsemigroup of (D_X,⋄).     

On the structure of a Morse form foliation

The foliation of a Morse form ω on a closed manifold M is considered. Its maximal components (cylinders formed by compact leaves) form the foliation graph; the cycle rank of this graph is calculated. The number of minimal and maximal components is estimated in terms of characteristics of M and ω. Conditions for the presence of minimal components and homologically non-trivial compact leaves are given in terms of rk ω and Sing ω. The set of the ranks of all forms defining a given foliation without minimal components is described. It is shown that if ω has more centers than conic singularities then b ₁(M) = 0 and thus the foliation has no minimal components and homologically non-trivial compact leaves, its folitation graph being a tree.      

On banach spaces which containl 1(τ) and types of measures on compact spaces

Two closely related results are presented, one of them concerned with the connection between topological and measure-theoretic properties of compact spaces, the other being a non-separable analogue of a result of Peŀczyński's about Banach spaces containingL ¹. Let τ be a regular cardinal satisfying the hypothesis that κ^ω<τ whenever κ<τ. The following are proved: 1) A compact spaceT carries a Radon measure which is homogeneous of type τ, if and only if there exists a continuous surjection ofT onto [0, 1]^τ. 2) A Banach spaceX has a subspace isomorphic tol ¹(τ) if and only ifX ^∗ has a subspace isomorphic toL ¹({0, 1}^τ). An example is given to show that a more recent result of Rosenthal's about Banach spaces containingl ¹ does not have an obvious transfinite analogue. A second example (answering a question of Rosenthal's) shows that there is a Banach spaceX which contains no copy ofl ¹ (ω₁), while the unit ball ofX ^∗ is not weakly^∗ sequentially compact.     

Subsequences of normal sequences

In this paper, we characterize a set of indices τ={τ(0)<τ(1)<…} such that forany normal sequence (α(0), α(1),…) of a certain type, the subsequence (α(τ(0)), α(τ(1)),…) is a normal sequence of the same type. Assume that_n→∞. Then, we prove that τ has this property if and only if the 0–1 sequence (θ _τ (0), whereθ _τ (i)=1 or 0 according asi∈{τ(j);j=0, 1,…} or not, iscompletely deterministic in the sense of B. Weiss.     

Semigroups with the Congruence Extension Property

ρT on a semigroup of T of S extends to the semigroup S, if there exists a congruence _ρ on s such that _{ρ|T= ρT}. A semigroup S has the congruence extension property, CEP, if each congruence on each semigroup extends to S. In this paper we characterize the semigroups with CEP by a set of conditions on their structure (by this we answer a problem put forward in [1]). In particular, every such semigroup is a semilattice of nil extensions of rectangular groups.     

Matrices of bisimple regular semigroups

A semigroup S is a matrix of subsemigroups S_iμ, i ε I, μ ε M if the S_iμ form a partition of S and S_iμS_jν≤S_iν for all i, j in I, μ, ν in M. If all the S_iμ are bisimple regular semigroups, then S is a bisimple regular semigroup. Properties of S are considered when the S_iμ are bisimple and regular; for example, if S is orthodox then each element of S has an inverse in every component S_iμ.     

O-simple strictly regular semigroups

In the previous paper [6], it has been proved that a semigroup S is strictly regular if and only if S is isomorphic to a quasi-direct product EX Λ of a band E and an inverse semigroup Λ. The main purpose of this paper is to present the following results and some relevant matters: (1) A quasi-direct product EX Λ of a band E and an inverse semigroup Λ is simple [bisimple] if and only if Λ is simple [bisimple], and (2) in case where EX Λ has a zero element, EX Λ is O-simple [O-bisimple] if and only if Λ is O-simple [O-bisimple]. Any notation and terminology should be referred to [1], [5] and [6], unless otherwise stated.     

Rational and transcendental growth series for the higher Heisenberg groups

This paper considers growth series of 2-step nilpotent groups with infinite cyclic derived subgroup. Every such group G has a subgroup of finite index of the form H _n ×ℤ ^m , where H _n is the discrete Heisenberg group of length 2n+1. We call n the Heisenberg rank of G. We show that every group of this type has some finite generating set such that the corresponding growth series is rational. On the other hand, we prove that if G has Heisenberg rank n ≧ 2, then G possesses a finite generating set such that the corresponding growth series is a transcendental power series. Oblatum 1-III-1995 & 28-XII-1995     

Semigroups generated by a group and an idempotent

It is well known that the semigroup of all transformations on a finite set X of order n is generated by its group of units, the symmetric group, and any idempotent of rank n ? 1. Similarly, the symmetric inverse semigroup on X is generated by its group of units and any idempotent of rank n ? 1 while the analogous result is true for the semigroup of all n × n matrices over a field.

In this paper we begin a systematic study of the structure of a semigroup S generated by its group G of units and an idempotent ? . The first section consists of preliminaries while the second contains some general results which provide the setting for those which follow.

In the third section we shall investigate the situation where G is a permutation group on a set X of order n and ? is an idempotent of rank n ? 1. In particular, we shall show that any such semigroup S is regular. Furthermore we shall determine when S is an inverse or orthodox semigroup or completely regular semigroup.

The fourth section deals with a special case, that in which G is cyclic. The fifth, and last, deals with the situation where G is dihedral. In both cases, the resulting semigroup has a particularly delicate structure which is of interest in its own right. Both situations are replete with interesting combinatorial gems.

The author was led to the results of this paper by considering the output of a computer program he was writing for generating and analyzing semigroups.     

P-systems in regular semigroups

In this paper, firstly it is shown that a regular semigroup S becomes a regular *-semigroup (in the sense of [1]) if and only if S has a certain subset called a p-system. Secondly, all the normal *-bands are completely described in terms of rectangular *-bands (square bands) and transitive systems of homomorphisms of rectangular *-bands. Further, it is shown that an orthodox semigroup S becomes a regular *-semigroup if there is a p-system F of the band E_S of idempotents of S such that F∋e, E_S∋t, e≥t imply t∈F. By using this result, it is also shown that F is a p-system of a generalized inverse semigroup S if and only if F is a p-system of F_S. Dedicated to Professor L. M. Gluskin on his 60th birthday     

Topological groups and convex sets homeomorphic to non-separable Hilbert spaces

Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover of X there is a sequence of maps (f _n : X → X) _nεgw such that each f _n is -near to the identity map of X and the family {f _n (X)} _n∈ω is locally finite in X. Also we show that a metrizable space X of density dens(X) < is a Hilbert manifold if X has gw-LFAP and each closed subset A ⊂ X of density dens(A) < dens(X) is a Z _∞-set in X.