数理解析研究所講究録 1222 巻 2001 年 86-89

86
Finitely Generated Idempotent-freeSemilattice-Indecomposable Semigroups with Relations I$\iota \mathrm{f}]$$f\backslash \uparrow\#$(Takayuki Tamura)$;’\overline{\tau}$

Department of Mathematics
University of California, DavisA semigroup$S$is called$S$-indecomposable if

image except the trivial semilattice. We assumeby$S\backslash \cdot \mathrm{S}^{\tau 2}$. Let$B=S\backslash \cdot 5^{2}’=\{a_{1}\ldots.,a_{k}\}$structure of idempotent-ffae$S$relation as defined below. Let

has no semilattice homomorphic$S$$S\neq \mathrm{b}^{\mathrm{v}2}$,$|S\backslash S^{2}|<\infty$

(1)

$S$

is generated. The purpose of this paper is to report the-indecomposable semigroup$Z_{+}$

and$S$generated by

$B$

with

be the set of aU positive integers. We assume

for some$a_{1}^{m_{1}}=\cdots=a_{k}^{m\iota}$$rn_{1}$,$\ldots$,$\sqrt k\in Z+\cdot$

In particular we study here the free semigroup satisfying (1), that is, every such semigroup is ahomomorphic image of the ffae one. The condition (1) is so strong that theproperty of$S-\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{e}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{h}.\mathrm{t}\mathrm{y}$Lemma 1. If$S$

is derived from (1).

is a semigroup generated by$B$

and satisfies (1),$d\iota\alpha\iota S$

isS-indecomposable.

Since $|B|<\infty$ the condition (1) is equivalent to $(1’)$ below.
such that
there exist ;4,For each pair ,$(1^{t})$$\alpha.$$n_{\mathrm{j}}\in Z_{+}$$a_{\dot{f}}\in B$$a^{n}.\cdot=a_{\dot{f}}^{\mathfrak{n}_{j}}$.

is powerjoimly generated by .
satisfies (1), equivalently (1), we saywith
be an idempotent-free semigroup which is power-jointly generated byLet. There is ahomomorphism : $Farrow S$(1), and let $F$ be the free semigroup overIf$B$

$S$

$B$

$B$

$S$

$f$

$B$

which satisfies the following conditions,i)$X\in F$,\"u)$f(a_{1})^{m_{1}}=\cdots=f(a_{k})^{m_{l}}$Let$\rho$$a\in B$$f(X)=f\{(a)\}\Rightarrow X=\{a\}$.

.

denote the congruence on $F$ generated by the set of binary relation$\{(a_{}^{m}‘,a_{j}^{m_{\dot{g}}}) :a:,a_{j}\in B\}$.

$\mathrm{s}$87
Then$S$is ahomomorphic image of

paper we study$F/\rho$

only if $X=Y$ inkeeping every element of$F/\rho$. For simplicity of notation, let $S=F/\rho$ , so.

fixed. In this$B$

in$X\rho Y$

if and$F$$‘ g^{\mathrm{Y}}$Prom (1) we immediately have
Lemma 2.

$a^{\lambda m_{l}}\dot{.}a_{j}^{x}=a_{j}^{x}a^{\lambda m_{l}}\dot{.}$for $i,j=1$ ,Let $X\in S$. $X$ has the form
(2)$a_{\mathrm{j}}\dot{.}\neq a_{\mathrm{j}+1}.\cdot$, , for any$k$$\ldots$$X=a_{1}^{x_{1}’}.\cdot\cdots$

where$a_{s}^{x_{\acute{S}}}.\cdot$$\lambda\in Z_{+}$$\mathrm{A}_{\dot{g}}.\in B$.

, $(j=1, \ldots, s)x’\dot{.}\in Z_{+}$ ,, $(j=1, \ldots, s-1)$ .

: where $0<x:\leq m:$ , $\lambda_{:}\in Z_{+}^{0}=Z_{+}\cup\{0\}$ . Let$M=a_{1}^{m_{1}}=\cdots=a_{k}^{m_{k}}$ .
By using Lemma 2repeatedly we have
We rewrite(3)$x’\dot{.}=x:+\lambda:m$$X=a_{1}^{x_{1}}.\cdot\cdots a^{x}\dot{.}.\cdot a_{\dot{1}1}^{\lambda_{1}m:_{1}}\ldots$

Likewise$Y$$=a_{j_{1}}^{y1}\cdots$$a_{j\iota}^{y\iota}a_{j_{1}}^{\mu_{1}m_{j_{1}}}\cdots$$a^{\lambda.m}.\cdot.‘\cdot=a_{1}^{x_{1}}\dot{.}\cdots$$a_{j\ell}^{\mu_{\ell}m_{\mathrm{j}_{l}}}=a_{j_{1}}^{y1}\cdots$$a^{x}.\cdot.\cdot M^{\lambda}$$a_{\mathrm{j}\iota}^{y\iota}M^{\mu}$

.

where

$\lambda=\lambda_{1}+\cdots+\lambda_{\epsilon}$

where $\mu=\mu_{1}+\cdots+\mu_{t}$ .

Consider the product $XY$. Again by using Lemma 2we have:If$i_{s}\neq j_{1}$$XY=a_{1}^{x_{1}}\dot{.}\cdots$

If $i_{s}=j_{1}$ andwhereIf$x_{s}+y_{1}\leq 2\mathrm{r}\mathrm{n},.$. andand. and$0<z_{s}\leq rr\mathit{4}.$$i_{s}=j_{1}$$P$satisfying, then$XY=a_{1}^{x_{1}}\dot{.}\cdots$, then$z_{s}\equiv x_{s}+y1(\mathrm{m}\mathrm{o}\mathrm{d} m\dot{.}.)$,$j=1$ ,$\ldots$$a_{-1}^{x_{-1}}\dot{.}.\cdot a^{z}.\cdot.\cdot a_{j_{2}}^{y2}\cdots$$a_{*\cdot-1}^{x.-1}.a_{\dot{1}}^{Z}.\cdot a_{j_{2}}^{y2}\cdots$$a_{j_{l}}^{\mathrm{V}t}M^{\lambda+\mu+1}$.

of elements of$V$$a_{j\iota}^{y\iota}M^{\lambda+\mu}$.

$XY=a_{1}^{x_{1}}.\cdot\cdots$

denote the set of finite sequences$a_{\mathrm{j}}\dot{.}\neq a_{i_{\mathrm{j}-1}}$.

$z_{s}\equiv x_{s}+8/1(\mathrm{m}\mathrm{o}\mathrm{d} 7n:.)$$x_{s}+y_{1}>2\mathrm{r}\mathrm{n},.$where $0<z_{s}\leq 7n_{1}$Let$a^{x}\dot{.}.\cdot a_{j_{1}}^{y1}\cdots$ $a_{j_{\ell}}^{y\iota}M^{\lambda+\mu}$$B$, $V=a$:if$i_{s}\neq j_{1}$if$i_{s}=j_{1}$$\cdots$$a.\cdot$, $s-1$ .

.

The binary operation on $P$ is defined by$a$:$\cdots$$a.\cdot.a_{j_{1}}\cdots$$a_{\mathrm{j}}$$a$:$\cdots$$a\dot{.}.a_{j_{2}}\cdots$$a_{\mathrm{j}_{C}}$$(a:_{1}\cdots\alpha..)*(a_{j_{1}}\cdots a_{j_{l}})=\{$

that is, if$i_{s}\neq j_{1}$, the product is juxtaposition, if $i_{s}=j_{1}$ then one of$a.\cdot$. and$a_{j_{1}}$isomitted.

Proposition 1. $P$ is a semigroup and$S$is homomorphic onto$P$

under the mapping$a_{1}^{x_{1}}\dot{.}\cdots a_{\iota}^{X_{l}}\dot{.}arrow a:_{1}\cdots a\dot{\ldots}$$P$

is regarded as the set of finite sequences

subject to$i_{j}\neq i_{j+1}$, $j=1$ ,$\ldots$, $s-1$ ,$s\geq 1$$i_{1}\cdots$$i_{\epsilon}$of elements of. In the form (3):$B=\{1, \ldots, k\}$

$X=a_{1}^{x_{1}}\dot{.}\cdots$$a_{}^{ax}.\cdot M^{\lambda}$, we88
rewriteby $x_{f}\dot{.}(j=1, \ldots, s)$$x_{j}$.

$(3’)X=a_{_{1}}^{oe\mathrm{s}_{1}}\cdots a^{x}.\cdot.\cdot..M^{\lambda}$

of$\{$,$\ldots$

and$r\mathrm{h}_{f}.)$$j.\ldots$

1,.

$k\}$

$x$

$\cdots$$i_{1}\cdots$$x$$j_{\epsilon}arrow y_{\mathrm{j}_{1}}\cdots$ $y_{j}$on$\varphi\psi$. and let$\Phi$$\cdots$, $j=1$ ,$0<x_{f}.\cdot\leq r\mathrm{h}_{f}.$operation.

of elements$x:$
is regarded as amapping from asequence:(i.e. an element modulo
such thatto asequence :The sequence$\ldots$$X:$.

$x_{\mathrm{j}}.\cdot\in Z_{m_{l_{j}}}$, ,$s=1$ ,

$s$$\ldots$,

$k$

. Let

$:_{C}\neq j_{1}$,$(i_{1}\cdots i_{\epsilon})*(j_{1}\cdots j_{t})=i_{1}\cdots$

If$i_{\epsilon}=j_{1}$,$(i_{1}\cdots i_{\epsilon})*(j_{1}\cdots j_{t})=i_{1}\ldots$ $i_{-1}.i_{\epsilon}j_{2}\cdots$ $j_{t}arrow x$.

$+y_{\dot{J}1}(\mathrm{m}\mathrm{o}\mathrm{d}_{7}n_{t}.)$Proposition 2.

$\Phi$,$0<z_{\dot{1}}$.

$\leq\pi_{4}..$

where

Define amapping $g:\Phi$$\mathrm{x}\Phi${$(\varphi,$$\lambda)$:$\varphi\in\Phi$, A$i_{\epsilon}j_{1}\cdots$:$j_{t}arrow x$$S$:$\ldots$$x.\cdot.$$g$.

$x.\cdot\cdot-1z.\cdot.y_{\dot{\infty}}\cdots$$y_{\mathrm{j}_{l}}$, where$\Phi$

under the mapping.

and$x\dot{.}$$\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{i}\mathrm{f}i_{\epsilon}=j_{1}$.

$+y_{\dot{\mathrm{J}}1}>r\mathrm{b}.$.

} and define the binary operation on$(\varphi,\lambda)(\psi,\mu)=(\varphi\psi/, \lambda+\mu+g(\varphi,\psi))$

Note that$y_{j}‘$

as follows:$arrow Z_{+}^{0}$$\in Z_{+}^{0}$$\cdots$ $x.\cdot.y_{1}\cdots$

is homomorphic onto$\varphi:i_{1}\cdots$ $i_{\epsilon}arrow x:1\ldots$$g(\varphi,\psi)=\{01$$\Gamma=$$\psi$.

is a semigroup, and$X=a_{1}^{\Leftrightarrow 1}.\cdot‘\cdots$ $a_{\dot{1}}^{x}.\cdot..M^{\lambda}arrow\varphi$

Let$\varphi’ \mathrm{s}$$\cdots$

as follows:

If$z_{4}\equiv x.\cdot$:$X:.$ ,:and define the binary$\varphi:i_{1}\cdots$ $i_{\epsilon}arrow x$

denote the set of all such$\Phi$$i_{\epsilon}$

as follows:$\Gamma$.

satisfies the condition:
for au$g(\varphi,\psi)+g(\varphi\psi/,\xi)=g(\varphi,\psi\xi)+g(\psi,\xi)$$\varphi,\psi,\xi\in\Phi$.

Now we have the main theorem

Theorem.

$\Gamma$

is a $se\sqrt.gmup$ and$a_{1}^{x_{l_{1}}}.\cdot\cdots a_{\dot{1}}^{x}.‘.M^{\lambda}arrow(\varphi,\lambda)$

whereThe idea of constructing

$S$

$S$

is isomorphic onto:$\varphi:i_{1}\cdots:_{C}arrow x$$\cdots$$x.\cdot.$$\Gamma$under the mapping $X=$.

-indecomposable semigroups from acertain free semigroup

was initiated by the author in case of finite nil semigroups 1958 [2], and also the idea wasused in case of finitely generated
The representation of

$S$

$Z$

semigroups [3].

by means of$\Gamma$

is similar to $N$ semigroups (i.e. idempotent-ffoe cancellative commutative archimedean semigroups) [1]89
Acknowledgement . The author expresses heartfelt appreciation to Professor Kunitaka
Shoji for advice and presentation on behalf of tlie autlior at the Research $C/onfem\iota ce$ .

REFERBNCBS[1] Tamura, T. Commutative nonpotent archimedean semigroups with cancellation law , Journal of$\mathrm{I}$Gakugei, Tokushima University,$\mathrm{V}\mathrm{o}\mathrm{l}$VIII, (1957), 5-11.

[2] Tamura, T. The theory of construction of finite semigroups.

III.

Osaka Mathematical Journal

10(1958) 191-204.
[3] Tamura, T. Note on$\mathrm{S}-$indecomposal le semigroups having zero as aunique idempotent$\mathrm{I}\mathrm{I}$, toappear in Proceedings of the Fourth Symposium on Algebra, Languages and Computation, Japan