数理解析研究所講究録 1222 巻 2001 年 1-4

1

Some remarks on generalized inverse*-semigroups II 1
Teruo Imaoka (Shimane University)
今岡 輝男 (島根大学総合理工学部)

Abstract-semigroups[2], we
By using aconcept of representations of generalized-semigroup. The purpose
introduce anew partial product on ageneralizedof this paper is to give acharacterization of prehomomorphisms of generalized inverse-semigroups.
$\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}*$$\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}*$$*$1Introduction
Asemigroupsatisfies$S$

with aunary operation $*:Sarrow S$ is called a $regular*semigroup$ if it(i) $(x^{*})^{*}=x$ ;(ii) (xy)’$=y^{*}x^{*}$(iii) $xx^{*}x=x$ .

;-semigroup. An idempotent in is called aprojection if $e^{*}=e$ . For
Let be a
asubset of , denote the sets of idempotents and projections of by $E(A)$ and $P(A)$ ,respectively.
$S$

$e$

$\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}*$

$A$

Let

$S$

$S$

$A$

$S$

be a$\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}*$-semigroup. Define arelation$a\leq b$$\Leftrightarrow$$a=eb=bf$$\leq \mathrm{o}\mathrm{n}$for some$e$$S$

as follows:, $f\in P(S)$ .

Ais caUed ageneralized $inverse*$ -semigroup if $E(S)$ satisfies theidentity $xyzw=xzyw$ . In this case, $E(S)$ forms aband.
$\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}*\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}$$S$

Result 1.1. [1] Let a and b be elements of S. Then the following conditions are equivalent:(1) $a\leq b$ ,(2) $aa^{*}=ba^{*}$ and $a^{*}a=b^{*}a$ ,(3) $aa^{*}=ab$ ’ and $a^{*}a=a^{*}b$ ,(4) $a=aa^{*}b=ba^{*}a$ .
Moreover, if is a generalized $inverse*$ -semigroup, the conditions above are equivalent
to the following:$S$

lThis paper is an abstract and the details will be published elsewhere2for some(5) $a=eb=bf$$e$, $f\in E(S)$ .

-semigroup, defined above, is a partial orResult 1.2. [1] The relation $\leq on$ aMoreover, if
is a gene ralized inverseunaryoperation.
which preserves theder oncompatible.
-semigrvup,$\mathit{7}wular$

$*$

$S$

$S$

$\leq is$$*$We call the partial order , defined above, the natural order on .
-semigroups. Amapping : $Sarrow T$ is called aprehomomorLet and bephism, if it satisfies$S$$\leq$

$T$

$S$

$\phi$$\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}*$(i) (ab)(/) $\leq(a\phi)(b\phi)$ ,(ii) $(a\phi)^{*}=a^{*}\phi$,

for all $a,b\in S$ .

Result 1.3. [1] Let be a prehomomorphism of asemigrvup T. Then we have the following:$\phi$$f\mathrm{W}^{uar*}$-sernigroup S to a $regular*-$maps an idempotent of S to an idempotent of T, and so it maps a projection of
to a projection of T,(1)$\phi$(2)$\phi$

is isotone, that is, aimplies$\leq b$$a\phi\leq b\phi$$S$,

As ageneralzation of the Preston-Vagner representations, we obtain arepresentation-semigroup[2]. Anon-empty set $X$ with an equivalence relationof ageneralzed
be atransitive -set. Asubset. Let
is called atransitive -set, and denoted by
if there exists at most one element of for eachof $X$ is called an -single subset of$x\sigmay(x,y\in A)$ implies $x=y$ . Denot the set of allequivalence class induced by , that is,, the symmetric inverse semigroup on
by T. Amapping in-single subsets of$X$ , is called apartial one-tO-One
are both -single subsets
ifmapping on
are the domain and the range of , respectively. Denote, whereandof.
bythe set of all partial one-t0-0ne -mappings ofby, define
For any -single subsets and of$\sigma$$\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}*$$A$$(X;\sigma)$$(X;\sigma)$$\iota$$\iota$$A$$(X;\sigma)$$\iota$$\sigma$$(X;\sigma)$$\iota$$\mathrm{I}_{X}$$\alpha$$(X;\sigma)$$\iota$$(X;\sigma)$$d(\alpha),r(\alpha)$$r(\alpha)$$d(\alpha)$$\alpha$$(X;\sigma)$$\iota$$B$$A$$\iota$$\iota$$(X;\sigma)$$\mathcal{G}\mathrm{I}(X_{j}\sigma)$$\theta_{A,B}$$\theta_{A,B}=\{(a,b)\in A\cross B : (a,b)\in\sigma\}=(A\mathrm{x}B)\cap\sigma$.

Since asubset of an -single subset is also an -single subset,, and letby, define
indexed set of one-t0-0ne partial functions. Now, define amultiplication$\alpha,\beta\in \mathrm{Q}\mathrm{X}(\mathrm{x},\mathrm{a})$$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}*\mathrm{o}\mathrm{n}\mathcal{G}\mathrm{I}(X_{j}\sigma)$$\mathcal{M}$$\theta_{\alpha,\beta}=\theta_{r(\alpha),d(\beta)}$$\theta_{\alpha\beta}$$=\{\theta_{\alpha\beta} :as follows:$\alpha$0$\beta=\alpha\theta_{\alpha,\beta}\beta$

1It is cffied $\mathrm{a}\vee prehomomorphism$ in [4]and. For any, anand aunary$\theta_{A,B}\in \mathcal{G}\mathrm{I}(X_{j}\sigma)$$\iota$$\iota$$\alpha^{*}=\alpha^{-1}$,\alpha,\beta\in \mathcal{L}\mathrm{I}(X_{j}\sigma)\}$$\circ$3
where the multiplication of the right side of the first equality is that ofby
. In this paper, we useor simply by.
than$(\mathcal{G}\mathrm{I}(X;\sigma), \circ, *)$$\mathcal{G}\mathrm{I}(X;\sigma)(\mathcal{M})$$\mathcal{G}\mathrm{I}(X;\sigma)$$\mathrm{I}_{X}$. Denote$\mathcal{G}\mathrm{I}(X\sigma)$

rather$\mathcal{G}\mathrm{I}_{(X;\sigma)}(\mathcal{M})$

Result 1.4. [2] For a transitive -set $(X;\sigma)$ , we have the follorving:$\iota$, defined above, is a generalized inverse -semigroup. haveover, any generalized inverse -semigroup can be embedded {up $to*$ -isomorphism)in.
on some transitive -set$The*$ -groupoid

(1)

$*$

$\mathcal{G}\mathrm{I}(X;\sigma)$$*$$\mathcal{G}\mathrm{I}(X;\sigma)$(2)$E(\mathcal{G}\mathrm{I}(X;\sigma))=\mathcal{M}$(3) If$(X;\sigma)$$\iota$and$P(\mathcal{L}\mathrm{I}(X;\sigma))=${$1_{A}$

is the identity relation on $X$ , then$\sigma$:$A$

is an -single subset of $(X;\sigma)$ }.
$\iota$is the symmetric inverse semigroup$\mathcal{G}\mathrm{I}(X;\sigma)$

on $X$ .

$\mathrm{I}_{X}$2Characterization of prehomomorphisms-semigroup. For any element $a\in S$ ,Let be ageneralzed$r(a)$, respectively. Define anew partial product . on as follows:and$S$$\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}*$$aa^{*}$

and$a^{*}a$by $d(a)$$\mathrm{S}$$a\cdot b=\{$abif $r(a)=d(a^{*}abb^{*})$ md $d(b)=r(a^{*}aW^{*})$
undefined otherwise

The partial product . is called arestricted product of .
$S$Lemma 2.1. Let(1)$a\cdot$(2) If$a$

and$b$be elements of a generalized $inverse*$ -semigroup$S$.

is defined if and only if $a^{*}a=a’ abb’ a’ a$ and $bb^{*}=bb’ a’ abb’$ .

$b$$a\cdot$$b$

is defined, then $d(a\cdot b)=d(a)$ and $r(a\cdot b)=r(b)$ .

The follwing lemma is abasic property of the restricted product of .
$S$

Lemma 2.2. Let

be a generalized $inverse*$ -semigroup.

$S$(1) Let be an element of and a projection of such that $e\leq x^{*}x$ . Then $a=xe$ isthe unique element in such that $a\leq x$ and $a^{*}a=e$ .
$S$

$x$

$S$$e$$S$(2) Let be an element of and a projection of such that $e\leq xx^{*}$ . Then $a=ex$ is
the unique element in such that $a\leq x$ and $aa^{*}=e$ .
$S$

$x$

$S$

$e$

$S$

(3) For any elemants $x,y\in S$ , $xy=a\cdot$$f=yy^{*}x^{*}xyy^{*}$$b$where $a=xe$ , $b=fy$ , $e=x^{*}xyy^{*}x^{*}x$ and.

Lemma 2.3. Let : $Sarrow T$ be a prehomomorphism of a generalized inverse -semigroup
to a generalized $inverse*$ -semigroup , and $a,b$ elements of .
$*$$\phi$$S$$T$(1) $(aa^{*})\phi=(a\phi)(a\phi)^{*}$ and $(a^{*}a)\phi=(a\phi)^{*}(a\phi)$ .

$S$

4(2) If a. b is defined, then$a\phi$.

$b\phi$is defined and $((a$ .

$b)\phi)^{*}(a\cdot b)\phi)=(a\phi\cdot b\phi)^{*}(a\phi\cdot b\phi)$.

(3) By the lemma above, $ab=(ae)\cdot$ $(fb)$ where $e=a’ abb’ a’ a$ and $f=bb’ a’ abb’$ . Ifsatisfies that $(gh)\phi=(g\phi)(h\phi)$ for any $g,h\in E(S)$ , then $(ae)\phi=(a\phi)(e\phi)$ and$\phi$$(fb)\phi=(f\phi)(b\phi)$.

Now we have the main theorem.
Theorem 2.4. Let

and$S$$T$be generalized inverse -sernigroups and $\phi:Sarrow T$ a mapping.
$*$is a prehomomorphism if and only if it preserves the restricred product and thenatural order.

(1)$\phi$(2)$\phi\dot{u}$a homomorphism if and only$(e\phi)(f\phi)$

for $dl$ , $f\in E(S)$ .

$\dot{\iota}f$

it is a prehomomorphism which satisfies (ef)(/)$=$$e$

References[1] Imaoka, T., Prehomomorphisms on $r\Psi ular*$ -semigroups, Mem. Fc. Sci. Shimane Univ.
15 (1981), 2327.

[2] Imaoka, T. and M. Katsura, Representations of locally $inverse*$ -semigrvups II, Semiyoup Forum 55 (1997), 247-255.
[3] Lawson, M. V., Inverse semigroups, World Scientific, Singapre, 1998.

[4] McAlister, D. B.,(1976), 215-231.

$\vee$-Prehomomorphisms on inverse semigroups, Pacific J. Math. 67

Department of Mathematics and Computer Science
Shimane University
Matsue, Shimane 690-8504, Japan-mail address: imaoka@math.shimane-u.ac.jp$e$