数理解析研究所講究録
910 巻 1995 年 87-88

87

Inverse semigroups and permutation properties
Kunitaka Shoji(庄司 邦孝)
Department of Mathematics, Shimane University

The purpose of this talk is to introduction a theorem on permutation properties of inverse semigroups,
and to give a comment on the proof of a lemma for the theorem.
appearing in Oknitski’s$\mathrm{b}\mathrm{o}\mathrm{o}\mathrm{k}[2]$Definition. A semigroup$S$

if there exists an integer $n\geq 2$ such thathas the permutation property.
for some$B_{n}$$\sigma\neq 1\in S_{n}$$w_{1}w_{2}\ldots w_{n}=w_{\sigma(1)}w_{\sigma(2)}\ldots w_{\sigma(n)}$Definition. A ring $R$ satifies a polynomial identity $p(x_{1}, x_{2}, \ldots x_{n})$ if all coefficients of $p(x_{1}, x_{2}, \ldots x_{n})$and $p(r_{1}, r_{2}, \ldots r_{n})=0$ for all $r_{i}\in R$ .
a PI-ring.
In this case, $R$ is$\mathrm{a}\mathrm{r}\mathrm{e}\pm 1$$\mathrm{c}\mathrm{a}\mathrm{U}\mathrm{e}\mathrm{d}$Problem [3, Restivo and Reutenauer].
Does the semigroup ring $k[S]$ of a semigroup
identity?

$S$

has the permutation property satisfy a polynomial

Theorem ([2, Theorem 23]). Let be an inverse semigroup. Then the following are eqivalent :(1) is finitely generated and satisfies the permutation property.
(2) has finitely many idempotents, and all subgroups of are finitely generated and abelian-by-finite.
(3) $K[S]$ is a left and right noetherian PI-algebra.
$S$

$S$

$S$

$S$

A proof of the theorem is based on Shirshov’s results concerning combinatorics on words, Blyth’sresults concerning groups with the permutation property, and structure theorems in semigroup ringtheory.
give a semigroup theoretical proof of the following lemma used for the proof of the theoremWe$\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{U}$above.

Lemma ([2, Lemma 22]). Let be a finitely generated inverse semigroup. If has the permutation
property, then has finitely many idempotents.
Proof. Let $a\in S$ . Then the principal factor semigroup $S_{a}=J_{a}/I(a)$ is a (ksimle (or simple) semigroupis a completely Osimple (or simple) semigroup. Here we assume that(see [1]). By [2, Theorem 17],
ofis less than, where is a positive integer. By [2, Proposition 19], the number ofhas$D$of has at most-classes. By the way of Sh\"uzenberger representation,. Thus, eachof lessis embedded in the direct product of row-monomial or column-monomial matrix semigroups
$S$

$S$

$S$

$S_{a}$$S$$B_{n}$$m= \frac{n}{2}$$n$$\mathcal{R}[\mathcal{L}]$$S$$S_{a}$$S$$m\mathcal{R}[\mathcal{L}]$$S_{i}$

88with zero. Then each has idempotent-separating congruence such thatthan $m$ over groupsof rank
is embedded in the direct product of row-monomial or column-monomial matrix semigroups$s\in
S,$So,idempotent.
isananyThenforwith zero.
less than $m$ over a singleelement groups
is periodic. Thus, by Restivo and Reutenauer’s result, $S/p$ has only finitely many idempotents, and$S$$G_{i}$$S/\rho$$\rho$$S_{i}$$s^{m!}$$\{e\}$$S/\rho$so does .
$S$

Remark. In the proof above, row-monomial matrix semigroups of rank $m$ over a singleelement groupare the partial transformation semigroups of a set of $m$ elements.

references[1] Clifford, A.H. &G.B. Preston, Algebraic thoery of semigroups, Vol.I, Mathematical Survey, No.7,
Amer. Math. Soc. Providence, R. I., 1961.

, Semigroup algebras, Marcel Dekker, 1991.
[2][3] Restivo A. and C. Reutenauer, On the Burnside problem for semigroups, J. Algebra 89(1984), 102-104.
$\mathrm{O}\mathrm{k}\mathrm{n}\dot{\acute{\mathrm{m}}}\mathrm{s}\mathrm{k}\mathrm{i}$