COUNTABLE VERSUS UNCOUNTABLE RANKS IN INFINITE SEMIGROUPS OF TRANSFORMATIONS AND RELATIONS

The relative rank $\rank(S:A)$ of a subset $A$ of a semigroup $S$ is the minimum cardinality of a set $B$ such that $\langle A\cup B\rangle=S$. It follows from a result of Sierpiński that, if $X$ is infinite, the relative rank of a subset of the full transformation semigroup $\mathcal{T}_{X}$ is either uncountable or at most $2$. A similar result holds for the semigroup $\mathcal{B}_{X}$ of binary relations on $X$.A subset $S$ of $\mathcal{T}_{\mathbb{N}}$ is dominated (by $U$) if there exists a countable subset $U$ of $\mathcal{T}_{\mathbb{N}}$ with the property that for each $\sigma$ in $S$ there exists $\mu$ in $U$ such that $i\sigma\le i\mu$ for all $i$ in $\mathbb{N}$. It is shown that every dominated subset of $\mathcal{T}_{\mathbb{N}}$ is of uncountable relative rank. As a consequence, the monoid of all contractions in $\mathcal{T}_{\mathbb{N}}$ (mappings $\alpha$ with the property that $|i\alpha-j\alpha|\le|i-j|$ for all $i$ and $j$) is of uncountable relative rank.It is shown (among other results) that $\rank(\mathcal{B}_{X}:\mathcal{T}_{X})=1$ and that $\rank(\mathcal{B}_{X}:\mathcal{I}_{X})=1$ (where $\mathcal{I}_{X}$ is the symmetric inverse semigroup on $X$). By contrast, if $\mathcal{S}_{X}$ is the symmetric group, $\rank(\mathcal{B}_{X}:\mathcal{S}_{X})=2$.AMS 2000 Mathematics subject classification: Primary 20M20

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

Let $\tau$ be a partition of the positive integer $n$. A partition of the set $\{1,2,\dots,n\}$ is said to be of type $\tau$ if the sizes of its classes form the partition $\tau$ of $n$. It is known that the semigroup $S(\tau)$, generated by all the transformations with kernels of type $\tau$, is idempotent generated. When $\tau$ 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 $S(\tau)$. We further develop existing techniques, associating with a subset $U$ of the set of all idempotents of $S(\tau)$ with kernels of type $\tau$ a directed graph $D(U)$; the directed graph $D(U)$ is strongly connected if and only if $U$ is a generating set for $S(\tau)$, a result which leads to a proof if the fact that the rank and the idempotent rank of $S(\tau)$ are both equal to$$ \max\biggl\{\binom{n}{d},\binom{n}{d+1}\biggr\}. $$AMS 2000 Mathematics subject classification: Primary 20M20; 05A18; 05A17; 05C20

NORMAL SEMIGROUPS OF ENDOMORPHISMS OF PROPER INDEPENDENCE ALGEBRAS ARE IDEMPOTENT GENERATED

Let $\mathcal{A}$ be a proper independence algebra of finite rank, let $G$ be the group of automorphisms of $\mathcal{A}$, let $a$ be a singular endomorphism and let $a^G$ be the semigroup generated by all the elements $g^{-1}ag$, where $g\in G$. The aim of this paper is to prove that $a^G$ is a semigroup generated by its own idempotents.AMS 2000 Mathematics subject classification: Primary 20M20; 20M10; 08A35