数理解析研究所講究録 1222 巻 2001 年 83-85

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Finite regular semigroups which areamalgamation bases for finite semigroups
庄司 邦孝 (Kunitaka Shoji)

島根大学総合理工学部

Department of Mathematics,Shimane University
Let $A$ be aclass of semigroups. Atriple of semigroups $S,T$, with$=S\cap T$ being asubsemigroup ofand is called an amalgam of and$A$
with acore in and denoted by $[S,T;U]$ . An amalgama $[S,T;U]$ of$A$ is weakly embedable in $A$ if there exist asemigroup $K$ belonging to $A$: $Tarrow K$ such that the restrictions: $Sarrow K$ ,and monomorphisms).
are equal to each other (that is,to of andIn the case that $\xi_{1}(S)\cap\xi_{2}(T)=\xi_{1}(U)$ , we say that an amalgama $[S,T;U]$
in $A$ is amalgarnationof $A$ is strongly embeddable in $A$. Asemigroup
in $A$
base [resp. weak amalgamation base] if any amalgam with acore$A$
is strongly embeddable [resp. weakly embeddable] in . In this paper, wesemigroups or therestrict ourselves to the cases that $A$ is the class of“amalgamationbase forterms
usetheWewill
class of aU finite semigroups.
semigroups’ or “weak amalgamation base forfinite semigroupi’ in the formercase or the latter.
. and Putcha [8] proved that any
semigroup is anmation base for ffi finite semigroups if the $J$-classes of $U$ is linearly orderedover has azero Jacobson radical. As aand the semigroup algebraanyfinite inverse semigroup is an amalobtained
that
they
by-products
gamation base for aU finite semigroups. In the paper [9] we gave aproof ofthe result by using representations of semigroups only. The purpose of thispaper is to show that the same method enable to extend the the reslut ffom
inverse semigroup to regular semigroups whose Rees factors satisfy the theconditions,and Anni
$U$

$U$

$S$

$T$

$S$

$U$

$T$

$\xi_{1}$$\xi_{1}$$U$$\xi_{1}(S)\cap\xi_{2}(T)\supseteq\xi_{1}(U)$$\xi_{2}$$\xi_{1}$

$U$

$U$

au

$\underline{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}}\mathrm{t}\mathrm{e}$$\mathrm{O}\mathrm{k}\mathrm{n}\dot{\acute{\mathrm{m}}}\mathrm{h}$$\mathbb{C}[U]$$U$$\mathrm{m}\infty \mathrm{a}-$$\mathbb{C}$$U$$Ann_{l}$Let be asemigroup with zero, 0, and $a,b\in S$ .
The set $\{s\in U| sa=0\}$ is called the left annihilator of in and isdenoted by $ann\iota(a)$ .
if $ann_{l}(a)=$satisfies the condition
In this case, we say thatanni(b) implies $aU=bU$ .
$U$

$a$

$U$

$Ann_{l}$$S$

84
The right annihilator and the condition $Ann_{r}$ axe defined by left-rightduality.
Theorem 1. Let be a finite regular semigroup with a chain of ideals
is a completely O-simple
is a maximal subgroup and eachsuch that$Ann_{r}(1\leqi\leq n-1)$ . Thensemigroups satisfying the conditions Anni andis an amalgamation base for finite semigorups.
$U$$U_{}/U_{+1}.\cdot$$U_{n}$$U$

Now we can present an example of asemigroup which does not satisfy theright annihilator condition but is an amalgamation base for finite semigroups
be the symmetric group of degree on the set $X=\{1,2, \cdots, n\}$Let
the right zero semigroup $X$ with multiplication $ij=j(i,j\in X)$ .
andwith multiplication of :Let).
$x=x$ ( $x\in R_{n}$ andimage of by andhas the representation extension property and the free repreThensentation extension property. Consequently, is an amalgamation base forsemigroups (see [3]).
$n$$S_{n}$$R_{n}$$U$$U=S_{n}\cup R_{\mathfrak{n}}$$x\cdot\alpha=\mathrm{t}\mathrm{h}\mathrm{e}$$x$$\alpha$$\alpha\in S_{\iota}.$$\alpha\cdot$

$U$

$U$

Theorem 2. The semigroup $U=S_{n}\cup R_{n}$ with multiplication of :$x\cdot\alpha=the$ image).
is anof by and $x=x$ ( $x\in R_{n}$ andamalgamation base for finite sernigrvups or not
$U$

$x$

$\alpha$$\alpha\cdot$$\alpha\in S_{n}$$U$References[1] S. Bulman-Fleming and K. McDowell, Absolutelyfiat semigroups. Pacific
J. Math. 107(1983), 319-333.
[2] A. H. Cxford and G. B. Preston, Algebraic theory of semigroups, Amer.
Math. Soc, Math. Survey, No.7, Providence, R.I., Vol.I(1961);
Vol.II(1967).

[3] T. E. Hall. Representation dension and amalgamation for semigroups.
Quart. J. Math. Oxford (2) 29(1978), 309-334.
[4] T. E. Hall. Finite inverse semigroups and amalgamations. Semigroups
and their applications, Reidel, 1987, pp. 51-56.

[5] T. E. Hall and M.S. Putcha, The potential $J$ -relation and amalgamation
bases forfinite semigrv ups, Proc. Amer. Math. Soc. 95(1985), 309-334.
[6] T. E. Hall and K. Shoji, Finite bands and amalgamation bases for finite
semigroups, Communications in algebras, To appear

85[7] B. H. Neumann, An essay on free products of groups
amalgamations
Trans. Roy. Soc. London Ser. A246(1954), 503-554.
$\ovalbox{\tt\small REJECT} th$[8] J. Oknifiski and M.S. Putcha, Embedding finite semigroup amalgams, J.
Austral. Math. Soc. (Series A) $51(1991)$ , 489-496.
[9] K. Shoji, A proof of Okninski and Putcha’s theorem, Sumitted