Exercise 14, § 2

For an associative law on a finite set $E$ there exist minimal subsets of the form $aE$ (that is minimal elements of the set of subsets of $E$ of this form, ordered by inclusion). (a) If $M = aE$ is minimal, then $xM = xE = M$ for all $x \in M$; with the law induced by that on $E$, $M$ is a left semi-group (Exercise 11) in which every left translation is bijective (cf. Exercise 13). (b) If $M = aE$ and $M' = a'E$ are minimal and distinct, then $M \cap M' = \varnothing$; for all $b \in M$, the mapping $x' \mapsto bx'$ of $M'$ into $M$ is a bijective mapping of $M'$ onto $M$. Deduce that there exist an idempotent $u' \in M'$ such that $bu' = b$ and an idempotent $u \in M$ such that $uu' = u$ (take $u$ to be the idempotent such that $bu = b$). Show that $u'u = u'$ (consider $uu'u$) and that every $y' \in M'$ such that $y'u = y'$ belongs to $M'u'$. (c) Show that the mapping $x' \mapsto ux'$ of $M'u'$ into $M$ is an isomorphism of $M'u'$ onto $Mu$; deduce that $M$ and $M'$ are isomorphic left semi-groups. (d) Let $M_i$ ($1 \leq i \leq r$) be the distinct minimal subsets of the form $aE$: deduce from (b) that the idempotents of each left semi-group $M_i$ can be arranged in a sequence $(u_{ij})$ ($1 \leq j \leq s$) so that $u_{ij}u_{kj} = u_{ij}$ for all $i, j, k$. If $K$ is the union of the $M_i$, show that $Eu_{ij} \subset K$ (note that, for all $x \in E$, $xu_{ij}E$ is a minimal subset of the form $aE$); deduce that $Eu_{ij}$ is the union of the $u_{kj}Eu_{kj}$ for $1 \leq k \leq r$ (show that $(Eu_{ij}) \cap M_k = u_{kj}Eu_{kj}$) and that $Eu_{ij}E = K$. Prove finally that every minimal subset of the form $Eb$ is identical with one of the $s$ sets $Eu_{ij}$ (note, using (a), that $Ebu_{ij}b = Eb$ and deduce that $Eb \subset K$).