西南大学学报 (自然科学版)  2019, Vol. 41 Issue (10): 37-44.  DOI: 10.13718/j.cnki.xdzk.2019.10.005
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  • 具有良恰当断面的富足半群的结构    [PDF全文]
    孔祥军1, 王蓓2     
    1. 曲阜师范大学 数学科学学院, 山东 曲阜 273165;
    2. 曲阜师范大学 软件学院, 山东 曲阜 273165
    摘要:利用${{\mathscr L}^ * }$-幂单半群和${{\mathscr R}^ * }$-幂单半群,给出具有良恰当断面的富足半群的一个对称的织积结构定理.此结论去掉了拟理想这个重要的前提,且比已有结论的形式更简单.其结果是对逆断面和恰当断面中相应结果的丰富和推广,为进一步研究该类半群的结构、性质及刻画其上的同余奠定了坚实的理论基础.
    关键词富足半群    良恰当断面    ${{\mathscr L}^ * }$-幂单半群    

    正则半群的逆断面[1]的概念于1982年引入.若正则半群S的逆子半群SoS的每个元素的唯一逆元,则称SoS的逆断面.在逆断面情形下,文献[2]引入了两个子半群RL,文献[3]证明了两个幂等子集IΛ都是带.作为逆断面的推广,恰当断面[4]的概念于1993年引入到富足半群中,文献[4]建立了具有可乘型A断面的富足半群的结构.文献[5]引入了两个幂等子集IΛ,并研究了恰当断面的若干性质.文献[6]引入并研究了两个重要子集RL,建立了具有拟理想恰当断面的富足半群的织积结构.文献[7]建立了具有左单S-恰当断面的富足半群的结构,文献[8]利用左正则带和${{\mathscr L}^ * }$-幂单半群,建立了具有S-恰当断面的富足半群的结构,但这两个半群是不对称的.文献[9]得到了完全正则半群簇的子簇的两种分解.文献[10]得到了变换半群的一个组合结果.文献[11-12]进一步讨论了恰当断面的性质,并研究了拟理想恰当断面的乘积问题.文献[13]对拟理想恰当断面的乘积问题进行了推广.文献[14]研究了可乘拟恰当断面的好同余.本文的主要目的是利用两个对称的半群${{\mathscr L}^ * }$-幂单半群和${{\mathscr R}^ * }$-幂单半群,建立具有良恰当断面的富足半群的结构.

    如果半群S的每一${{\mathscr L}^ * }$-类和${{\mathscr R}^ * }$-类都含幂等元,则称S为富足半群[15].幂等元集成带(半格)的富足半群称为拟恰当半群(恰当半群)[16].恰当半群的每一${{\mathscr L}^ * }$-类La*${{\mathscr R}^ * }$-类Ra*仅含一个幂等元,分别记作a*a+.用符号EEo分别表示半群SSo的幂等元集.

    U是富足半群S的富足子半群.如果对任意aU,存在两个幂等元eLa*(S)∩UfRa*(S)∩U,则称US的*-子半群.设So是富足半群S的*-恰当子半群.如果对任意xS,存在幂等元efE和唯一xSo,使得x=exf,这里$e{\overline {{\mathscr L}x} ^ + }$$f{\overline {{\mathscr R}x} ^ * }$,则称SoS的恰当断面.易知ef是由xSo唯一确定的,分别记为exfx,且有ex${{\mathscr R}^ * }$x${{\mathscr L}^ * }$fx.记

    $ I = \left\{ {{e_x}:x \in S} \right\}\;\;\;\mathit{\Lambda } = \left\{ {{f_x}:x \in S} \right\} $

    如果IΛ都是带,则称SoS的良恰当断面.据文献[5]的命题2.3知,若SoS的良恰当断面,则I是左正则带(iji=ij),Λ是右正则带(iji=ji).幂等元集成右正则带(左正则带)的拟恰当半群称为${{\mathscr L}^ * }$-幂单半群(${{\mathscr R}^ * }$-幂单半群).本文未定义的概念和符号见文献[6, 15-16].

    引理1[8]  设S是具有恰当断面So的富足半群.若I(Λ)是子半群,则R(L)也是子半群,从而R(L)是拟恰当半群.

    在下文中,R表示一个具有恰当断面So${{\mathscr R}^ * }$-幂单半群.则易知,对任一xR,有fx=x*Eox=exx.对${{\mathscr L}^ * }$-幂单半群L有对偶的结论.下面给出本文的主要定理.

    定理1  设${{\mathscr R}^ * }$-幂单半群R${{\mathscr L}^ * }$-幂单半群L具有一个共同的恰当断面So.对任一aL,设ϕaRR$y \mapsto {\phi _a}y$为映射;对任一xR,设ψxLL$b \mapsto b\psi _x$为映射.在集合U=R×L={(xa)∈R×Lx=a}上定义乘法

    $ \left( {x,a} \right)\left( {y,b} \right) = \left( {{e_x}\left( {{\phi _a}y} \right),\left( {a{\psi _y}} \right){f_b}} \right) $

    假设对任意abL和任意xyR,下列条件满足:

    (ⅰ) $\overline {{\phi _a}y} = \overline {a{\psi _y}} $

    (ⅱ)若x=b,则ϕa(ex(ϕby))=eϕax(ϕ(x)fby)且(ex(ϕby))fy=((x)fb)ψy

    (ⅲ)若aSo,则ϕay=ayy=ay;若xSo,则x=bxϕax=ax

    (ⅳ) ea(ϕax)=ϕax且(x)fx=x

    (ⅴ)对任一cLzR,若ϕay=ϕaz及(y)fb=(z)fc,则ϕfay=ϕfaz且(faψy)fb=(faψz)fc;若ey(ϕbx)=ez(ϕcx)及x=x,则ey(ϕbex)=ez(ϕcex)且ex=ex.

    U是具有同构于So的良恰当断面的富足半群.

    反之,每一个具有良恰当断面的富足半群都可以这样构造.

    定理1由下面几个引理逐步证得.

    引理2  U上的乘法是有定义的.

      只须证(ex(ϕay),(y)fb)∈U.

    ϕaRRψyLL是映射,显然有

    $ \left( {{e_x}\left( {{\phi _a}y} \right),\left( {a{\psi _y}} \right){f_b}} \right) \in R \times L $

    E(R)是左正则带,且

    $ {e_a}{e_x}{e_{{\phi _a}y}} = {e_a}{e_x}{e_{ea\left( {{\phi _a}y} \right)}} = {e_a}{e_x}{e_a}{e_{{\phi _a}y}} = {e_a}{e_{{\phi _a}y}} = {e_{{\phi _a}y}} $

    据条件(ⅰ)和(ⅳ),有

    $ {e_x}\left( {{\phi _a}y} \right) = {e_x}{e_{{\phi _a}y}}\overline {{\phi _a}y} $

    $ {e_x}{e_{{\phi _a}y}}\mathscr{L}{e_{{\phi _a}y}}\mathscr{L}{\overline {{\phi _a}y} ^ + } $

    $\overline {{e_{x\left( {{\phi _a}y} \right)}}} = \overline {{\phi _a}y} $.类似地,$\overline {\left( {a{\psi _y}} \right){f_b}} = \overline {a{\psi _y}} $.据条件(ⅳ),有

    $ \overline {{e_x}\left( {{\phi _a}y} \right)} = \overline {\left( {a{\psi _y}} \right){f_b}} $

    $ \left( {{e_x}\left( {{\phi _a}y} \right),\left( {a{\psi _y}} \right){f_b}} \right) \in U $

    引理3  U是半群.

      设(xa),(yb),(zc)∈U.则

    $ \begin{array}{l} [(x,a)(y,b)](z,c) = \left( {{e_x}\left( {{\phi _a}y} \right),\left( {a{\psi _y}} \right){f_b}} \right)(z,c) = \\ \;\;\;\;\;\;\;\;\;\left( {{e_{{e_x}\left( {{\phi _a}y} \right)}}\left( {{\phi _{\left( {a{\phi _y}} \right){f_b}}}z} \right),\left( {\left( {{\phi _a}y} \right){f_b}} \right){\psi _z}{f_c}} \right) = \\ \;\;\;\;\;\;\;\;\;\left( {{e_x}{e_{{\phi _a}y}}\left( {{\phi _{\left( {a{\psi _y}} \right){f_b}}}z} \right),\left( {\left( {{\phi _a}y} \right){f_b}} \right){\psi _z}{f_c}} \right) \end{array} $

    $ \begin{array}{l} (x,a)[(y,b)(z,c)] = (x,a)\left( {{e_y}\left( {{\phi _b}z} \right),\left( {b{\psi _z}} \right){f_c}} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{e_x}\left( {{\phi _a}\left( {{e_y}\left( {{\phi _b}z} \right)} \right)} \right),\left( {a{\psi _{{e_y}\left( {{\phi _b}z} \right)}}} \right){f_{\left( {b{\psi _z}} \right){f_c}}}} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{e_x}\left( {{\phi _a}\left( {{e_y}\left( {{\phi _b}z} \right)} \right)} \right),\left( {a{\psi _{{e_y}\left( {{\phi _b}z} \right)}}} \right){f_{b{\psi _z}}}{f_c}} \right) \end{array} $

    y=b,据条件(ⅱ),有

    $ [(x,a)(y,b)](z,c) = (x,a)[(y,b)(z,c)] $

    所以U是半群.

    引理4  设(xa)∈U.则(xa)∈E(U)当且仅当ϕax=x=a=x.

      因

    $ (x,a)(x,a) = \left( {{e_x}\left( {{\phi _a}x} \right),\left( {a{\psi _x}} \right){f_a}} \right) $

    注意到xRaL,易知:若

    $ {\phi _a}x = a{\psi _x} = \bar a = \bar x $

    $ \left( {{e_x}\left( {{\phi _a}x} \right),\left( {a{\psi _x}} \right){f_a}} \right) = \left( {{e_x}\bar x,\bar a{f_a}} \right) = (x,a) $

    故(xa)∈E(U).反之,若(xa)∈E(U),则ex(ϕax)=x且(x)fa=a.据(xa)∈UaL,有$ {e_x}{\overline {{\mathscr L}x} ^ + }$=a+=ea.故

    $ {\phi _a}x = {e_a}\left( {{\phi _a}x} \right) = {e_a}{e_x}\left( {{\phi _a}x} \right) = {e_a}x = {{\bar x}^ + }{e_x}\bar x = {{\bar x}^ + }\bar x = \bar x = \bar a $

    类似地,x=a=x.

    引理5  假设(xa)∈U,记u=(exx+)和v=(a*fa).则uvE(U)且$u{{\mathscr R}^ * }$(xa)${{\mathscr L}^ * }v$.

      据引理4,显然有uvE(U).令xEoaL,因为xfa=xfa=a,则

    $ \left( {{e_x},{{\bar x}^ + }} \right)(x,a) = \left( {{e_x}\left( {{\phi _{{{\bar x}^ + }}}x} \right),\left( {{{\bar x}^ + }{\psi _x}} \right){f_a}} \right) = \left( {{e_x}{{\bar x}^ + }x,\overline {{{\bar x}^ + }x} {f_a}} \right)\left( {x,\bar x{f_a}} \right) = (x,a) $

    假设(yb),(zc)∈Γ1,使得

    $ (y,b)(x,a) = (z,c)(x,a) $

    $ \left( {{e_y}\left( {{\phi _b}x} \right),\left( {b{\psi _x}} \right){f_a}} \right) = \left( {{e_z}\left( {{\phi _c}x} \right),\left( {c{\psi _x}} \right){f_a}} \right) $

    $ {e_y}\left( {{\phi _b}x} \right) = {e_z}\left( {{\phi _c}x} \right) $

    $ \left( {b{\psi _x}} \right){f_a} = \left( {c{\psi _x}} \right){f_a} $

    $ \left( {b{\psi _x}} \right){f_a}{f_x} = \left( {c{\psi _x}} \right){f_a}{f_x} $

    fa${{\mathscr R}^ * }$a*=x*=fx和条件(ⅳ),有x=x.因此据条件(ⅴ),有

    $ {e_y}\left( {{\phi _b}{e_x}} \right) = {e_z}\left( {{\phi _c}{e_x}} \right) $

    $ b{\psi _{{e_x}}} = c{\psi _{{e_x}}} $

    所以

    $ \left( {y,b} \right)\left( {{e_x},{{\bar x}^ + }} \right) = \left( {{e_y}\left( {{\phi _b}{e_x}} \right),\left( {b{\phi _{{e_x}}}} \right){{\bar x}^ + }} \right) = \left( {{e_z}\left( {{\phi _c}{e_x}} \right),\left( {c{\psi _{{e_x}}}} \right){{\bar x}^ + }} \right) = (z,c)\left( {{e_x},{{\bar x}^ + }} \right) $

    故(xa)${{\mathscr R}^ * }u$.对偶地,有(xa)${{\mathscr L}^ * }v$.

    引理6  U是富足半群.

      据引理5可得.

    引理7  设W={(ss):sSo}.则W是同构于SoU的恰当*-子半群.

      显然WU.设(ss),(tt)∈W.易知

    $ (s,s)(t,t) = \left( {{e_s}st,st{f_t}} \right) = (st,st) \in W $

    所以W是子半群.对任一sSo,定义=(ss),显然φ是一个同构.故So$ \cong $W.

    为证W是*-子半群,设(ss)∈W.据引理4和引理5,有

    $ u = \left( {{e_s},{s^ + }} \right) = \left( {{s^ + },{s^ + }} \right) \in E\left( W \right) $

    $u{{\mathscr R}^ * }$(ss).类似地,v=(s*s*)∈E(W)且$v{{\mathscr L}^ * }$(ss).

    引理8  设(x1a1),(x2a2)∈U.则:

    (ⅰ) (x1a1)${{\mathscr R}^ * }$(x2a2)当且仅当x1${{\mathscr R}^ * }$x2

    (ⅱ) (x1a1)${{\mathscr L}^ * }$(x2a2)当且仅当a1${{\mathscr L}^ * }$a2.

      为证(ⅰ),据引理5,只需证明(ex1${\overline {{x_1}} ^ + }$)${{\mathscr R}^ * }$(ex2${\overline {{x_2}} ^ + }$)当且仅当x1${{\mathscr R}^ * }$x2.

    x1${{\mathscr R}^ * }$x2,则ex1=ex2,故${\overline {{x_1}} ^ + }$=${\overline {{x_2}} ^ + }$.因此

    $ \left( {{e_{{x_1}}},{{\overline {{x_1}} }^ + }} \right) = \left( {{e_{{x_2}}},{{\overline {{x_2}} }^ + }} \right) $

    反之,若

    $ {u_1} = \left( {{e_{{x_1}}},{{\overline {{x_1}} }^ + }} \right){\mathscr{R}^*}{u_2} = \left( {{e_{{x_2}}},{{\overline {{x_2}} }^ + }} \right) $

    u1u2=u2u2u1=u1.故

    $ {e_{{x_1}}}\left( {{\phi _{{{\overline {{x_1}} }^ + }}} + {e_{{x_2}}}} \right) = {e_{{x_1}}}{\overline {{x_1}} ^ + } + {e_{{x_2}}} = {e_{{x_2}}} $

    $ {e_{{x_2}}}\left( {{\phi _{{{\overline {{x_2}} }^ + }}} + {e_{{x_1}}}} \right) = {e_{{x_2}}}{\overline {{x_2}} ^ + } + {e_{{x_1}}} = {e_{{x_1}}} $

    ex1ex2=ex2,且ex2ex1=ex1.故ex1${{\mathscr R}^ * }$ex2.所以x1${{\mathscr R}^ * }$x2.

    (ⅱ)可对偶地证明.

    引理9  WU的恰当断面.

      对任意的t=(xa)∈U,记

    $ \bar t = \left( {\bar x,\bar x} \right) \in W\quad {e_t} = \left( {{e_x},{{\bar x}^ + }} \right)\quad {f_t} = \left( {{{\bar a}^*},{f_a}} \right) $

    易验证t=ettft.而且

    $ {e_t}\mathscr{L}{{\bar t}^ + } = \left( {{{\bar x}^ + },{{\bar x}^ + }} \right)\;\;\;\;{f_t}\mathscr{R}{{\bar t}^*} = \left( {{{\bar a}^*},{{\bar a}^*}} \right) $

    假设t可以写成另一种形式t=ettft′,其中

    $ {{e'}_t} = \left( {{y_1},{b_1}} \right) \in E(U)\quad {{f'}_t} = \left( {{y_2},{b_2}} \right) \in E(U) $
    $ \bar t' = (\bar y,\bar y) \in W\;\;\;\;{{e'}_t}\mathscr{L}{{\bar t'}^ + } = \left( {{{\bar y}^ + },{{\bar y}^ + }} \right) $

    $ {{f'}_t} = {{\bar t'}^ * } = \left( {{{\bar y}^*},{{\bar y}^*}} \right) $

    根据引理8的证明,有

    $ \begin{array}{*{20}{c}} {{f_{{b_1}}} = {{\bar y}^ + }}&{{{\overline {{b_1}} }^*} = {{\bar y}^ + }}&{{{\bar y}^*} = {e_{{y_2}}}}&{{{\overline {{y_2}} }^ + } = {{\bar y}^*}} \end{array} $

    et${{\mathscr R}^ * }$t${{\mathscr R}^ * }$et′知

    $ \begin{array}{*{20}{c}} {{e_y} = {e_{{y_1}}}}&{{{\bar x}^ + } = {{\overline {{y_1}} }^ + }} \end{array} $

    类似地,有fa=fb2${\overline a ^ * } = {\overline {{b_2}} ^ * }$.所以,由b1Ly2R

    $ {b_1} = \overline {{b_1}} {f_{{b_1}}} = {\overline {{b_1}{b_1}} ^*} = \overline {{b_1}} \in {S^o} $
    $ {y_2} = {e_{{y_2}}}\overline {{y_2}} = \overline {{y_2}} + \overline {{y_2}} = \overline {{y_2}} \in {S^o} $

    从而

    $ \begin{array}{*{20}{c}} {{{e'}_t} = \left( {{y_1},\overline {{b_1}} } \right)}&{{{f'}_t} = \left( {\overline {{y_2}} ,{b_2}} \right)} \end{array} $

    et′,ft′∈E(U),根据引理4,有$\overline {{b_1}} {y_1} = \overline {{y_1}} $${b_2}\overline {{y_2}} = \overline {{b_2}} $.因y1R,有

    $ \begin{array}{*{20}{c}} {{y_1} = {e_{{y_1}}}\overline {{y_1}} }&{{{\overline {{y_1}} }^ + }{y_1} = {{\overline {{y_1}} }^ + }{e_{{y_1}}}\overline {{y_1}} = \overline {{y_1}} } \end{array} $

    y1${{\mathscr L}^ * }$$\overline {{y_1}} $${{\mathscr L}^ * }$$\overline {{y_1}}^* $.从$\overline {{b_1}} {y_1} = \overline {{y_1}} $可推断出$\overline {{y_1}} = \overline {{y_1}}{y_1} $.因$\overline {{b_1}} = \overline {{y_1}} $,从而$\overline {{y_1}} ^*= \overline {{y_1}} ^*{y_1}$,故

    $ {y_1}{y_1} = {y_1}{\overline {{y_1}} ^*}{y_1} = {y_1}{\overline {{y_1}} ^*} = {y_1} $

    因此y1是幂等元,且y1=ey1=ex.类似地,有

    $ {b_2} = {f_{{b_2}}} = {f_a} $

    易知

    $ {b_1} = \overline {{b_1}} = \overline {{y_1}} = \overline {{e_x}} = {{\bar x}^ + }\;\;\;\;{y_2} = \overline {{y_2}} = \overline {{b_2}} = \overline {{f_a}} = {{\bar a}^ * } $

    et′=etft′=ft.从而

    $ t = (x,a) = {{e'}_t}\bar t'{{f'}_t} = \left( {{e_x},{{\bar x}^ + }} \right)\left( {\bar y,\bar y} \right)\left( {{{\bar a}^*},{f_a}} \right) = \left( {{e_x}\bar y,{{\bar x}^ + }\bar y{f_a}} \right) $

    x=exy.由x=exxexy=exx,可得x+y=x.显然$\overline {{x}} ^+ = \overline {{b_1}}=\overline {{b_1}}^* =\overline {y}^+$,故x=y+y=y.所以,WU的恰当断面.

    引理10  WU的良恰当断面.

      设

    $ E\left( R \right)| \times |{E^o} = \left\{ {(x,a) \in U:{x^2} = x,a \in {E^o}} \right\} $
    $ {E^o}| \times |E(L) = \left\{ {(x,a) \in U:x \in {E^o},{a^2} = a} \right\} $

    对任一(xa)∈I(U),有(xa)∈E(U),且存在(yy)∈E(W),yEo,使得(xa)${{\mathscr L} }$(yy).则(a*fa)${{\mathscr L} }$(yy),故(a*fa)=(yy),即a*=y=fa.故aRL=So,根据引理4,有a=ϕax=ax.因(xa)∈U,有x=a=a,故x=xx.从引理9的证明可推断出x是幂等元,故x=ex.从而

    $ a = \bar a = \bar x = \overline {{e_x}} = {{\bar x}^ + } \in {E^o} $

    因此

    $ I(U) \subseteq E(R)| \times |{E^o} $

    对任一(xa)∈U,其中x2=xaEo,显然有

    $ (x,a) \in E(U)\;\;\;(x,a)\mathscr{L}(a,a) \in E(W) $

    所以I(U)=E(R)|×|Eo$ \cong $E(R)是左正则带.对偶地,Λ(U)=Eo|×|E(L)$ \cong $E(L)是右正则带.

    据定义知WU的良恰当断面.至此完成了定理1正面部分的证明.

    反之,假设S是富足半群,具有良恰当断面So.则I是左正则带,Λ是右正则带.从而,R${{\mathscr R}^ * }$-幂单半群,L${{\mathscr L}^ * }$-幂单半群,且RL分享一个共同的恰当断面So.对任一aLxR,设

    $ {\phi _a}:R \to R\;\;\;{\phi _a}x = ax{\overline {ax} ^*} $
    $ {\psi _x}:L \to L\;\;\;a{\psi _x} = {\overline {ax} ^ + }ax $

    为映射.则它们满足下列条件:

    (ⅰ)因

    $ {\phi _a}y = ay{\overline {ay} ^*} = {e_{ay}}\overline {ay} {f_{ay}}{\overline {ay} ^*} = {e_{ay}}\overline {ay} $

    ${e_{ay}}{\overline {{\mathscr L}ay} ^ + }$,有$\overline {{\phi _a}y} = \overline {a{_y}} $.类似地,因

    $ a{\psi _y} = {\overline {ay} ^ + }ay = \overline {ay} {f_{ay}} $

    ${f_{ay}}{\overline {{\mathscr R}ay} ^ * }$,有$\overline {a{\psi _y}} = \overline {ay} $.故$\overline {{\phi _a}y} = \overline {a{\psi _y}} $.

    (ⅱ)由x=bxRbL

    $ {e_x}b = {e_x}\bar b{f_b} = {e_x}\bar x{f_b} = x{f_b} $

    计算得

    $ \begin{array}{l} {\phi _a}\left( {{e_x}\left( {{\phi _b}y} \right)} \right) = {\phi _a}\left( {{e_x}\left( {by{{\overline {by} }^*}} \right)} \right) = a{e_x}\left( {by{{\overline {by} }^*}} \right){\overline {a{e_x}\left( {by{{\overline {by} }^*}} \right)} ^*} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{e_{a{e_x}\left( {by{{\overline {by} }^*}} \right)}},\overline {a{e_x}\left( {by{{\overline {by} }^*}} \right)} = {e_{a{e_x}by}}\overline {a{e_x}by} = {e_{ax{f_b}y}}\overline {ax{f_b}y} \end{array} $
    $ \begin{array}{*{20}{l}} {{e_{{\phi _{ax}}}}\left( {{\phi _{\left( {a{\psi _x}} \right){f_b}}}y} \right) = {e_{ax{{\overline {ax} }^*}}}\left( {{\phi _{\left( {{{\overline {ax} }^ + }ax} \right){f_b}}}y} \right) = {e_{ax}}{e_{\left( {{{\overline {ax} }^ + }ax} \right){f_b}y}}\overline {\left( {{{\overline {ax} }^ + }ax} \right){f_b}y} = }\\ {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{e_{ax}}{e_{ax{f_b}y}}\overline {ax{f_b}y} = {e_{ax{f_b}y}}\overline {ax{f_b}y} } \end{array} $

    $ {\phi _a}\left( {{e_x}\left( {{\phi _b}y} \right)} \right) = {e_{{\phi _{ax}}}}\left( {{\phi _{\left( {a{\psi _x}} \right){f_b}}}y} \right) $

    类似地,有

    $ \left( {a{\psi _{{e_x}\left( {{\phi _b}y} \right)}}} \right){f_{b{\psi _y}}} = \overline {ax{f_b}y} {f_{ax{f_b}y}} = \left( {\left( {a{\psi _x}} \right){f_b}} \right){\psi _y} $

    (ⅲ)若aSo,则aySoRR,故fay=ay*.所以

    $ {\phi _a}y = ay{\overline {ay} ^*} = ay{f_{ay}} = ay $
    $ a{\psi _y} = {\overline {ay} ^ + }ay = \overline {ay} {f_{ay}} = {\overline {ayay} ^*} = \overline {ay} $

    对偶地,若xSo,则x=bx,且ϕax=ax.

    (ⅳ)易知

    $ {e_a}\left( {{\phi _a}y} \right) = {e_a}ay{\overline {ay} ^*} = ay{\overline {ay} ^*} = {\phi _a}y $
    $ \left( {a{\psi _y}} \right){f_y} = {\overline {ay} ^ + }ay{f_y} = {\overline {ay} ^ + }ay = a{\psi _y} $

    (ⅴ)最后证明:若x=xey(ϕbx)=ez(ϕcx),则

    $ b{\psi _{{e_x}}} = c{\psi _{{e_x}}}\;\;\;\;{e_y}\left( {{\phi _b}{e_x}} \right) = {e_z}\left( {{\phi _c}{e_x}} \right) $

    ebx=ebex,由Eo是半格,有$\overline {{bx}}^+ = \overline {{be_{x}}}^+ $.类似地,$\overline {{cx}}^+ = \overline {{ce_{x}}}^+ $.从x=x可推断出

    $ {\overline {b{e_x}} ^ + } \cdot b{e_x}\bar x = {\overline {c{e_x}} ^ + }c{e_x}\bar x $

    所以$ {\overline {b{e_x}} ^ + }b{e_x} = {\overline {c{e_x}} ^ + }c{e_x}$,从而x=x.若ey(ϕbx)=ez(ϕcx),则eyebxbx =ezecxcx.因x=x,有$\overline {b{\psi _x}} = \overline {c{\psi _x}} $,据(ⅰ),bx =cx.从而

    $ {e_y}{e_{bx}}{\overline {bx} ^ + } = {e_z}{e_{cx}}{\overline {cx} ^ + } $

    eyebx=ezecx.所以eyebex=ezecex.类似地,因ex=ex,有$\overline {b{e_x}} = \overline {c{e_x}} $,故

    $ {e_y}{e_{b{e_x}}}\overline {b{e_x}} = {e_z}{e_{c{e_x}}}\overline {c{e_x}} $

    从而

    $ {e_y}b{e_x}{\overline {b{e_x}} ^ * } = {e_z}c{e_x}{\overline {c{e_x}} ^ * } $

    $ {e_y}\left( {{\phi _b}{e_x}} \right) = {e_z}\left( {{\phi _c}{e_x}} \right) $

    对偶地,若ϕay=ϕaz及(y)fb=(z)fc,则

    $ {\phi _{{f_a}}}y = {\phi _{{f_a}}}z\;\;\;\left( {{f_a}{\psi _y}} \right){f_b} = \left( {{f_a}{\psi _z}} \right){f_c} $

    所以得到一个富足半群U.最后证明U同构于S.

    设(xa)∈U.定义θUSθ((xa))=exa,则θ是有定义的,且θ是单射.事实上,若

    $ \bar x = \bar a\;\;\;\bar y = \bar b\;\;\;{e_x}a = {e_y}b $

    其中xyRabL.则从exa=exafa$f_a{\overline {{\mathscr R}a} ^ * } $$e_x{\overline {{\mathscr L}x} ^ + } = {\overline a ^ + }$可推断出$\overline {{e_x}a} = \overline a $.类似地,$\overline {{e_y}b} = \overline b$,故a=b.从而

    $ \bar x = \bar y\;\;\;\;{f_x} = {{\bar x}^*} = {{\bar y}^*} = {f_y} $

    所以

    $ x = {e_x}\bar x{f_x} = {e_x}\bar a{f_x} = {e_x}a{f_x} = {e_y}b{f_x} = {e_y}b{f_y} = {e_y}\bar b{f_y} = {e_y}\bar y{f_y} = y $

    类似地,a=b.

    对任意(xa),(yb)∈U,因yfb=eyb,有

    $ \begin{array}{l} \theta [(x,a)(y,b)] = \theta \left( {\left( {{e_x}ay{{\overline {ay} }^*},{{\overline {ay} }^ + }ay{f_b}} \right)} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{e_{e{x^{ay{{\overline {ay} }^*}}}}} \cdot {\overline {ay} ^ + }ay{f_b} = {e_x}{e_{ay{{\overline {ay} }^*}}} \cdot {\overline {ay} ^ + }ay{f_b} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{e_x}{e_{ay}}{\overline {ay} ^ + }ay{f_b} = {e_x}ay{f_b} = {e_x}a \cdot {e_y}b = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\theta ((x,a)) \cdot \theta ((y,b)) \end{array} $

    θ是同态.对任一xS,易验证xx*Rx+xL.而且,从

    $ x{{\bar x}^*} = {e_x}\bar x{f_x}{{\bar x}^*} = {e_x}\bar x\;\;\;{{\bar x}^*}\;\;\;{e_x}{\mathscr L}{{\bar x}^ + } $

    $ {{\bar x}^ + }x = {{\bar x}^ + }{e_x}\bar x{f_x} = {{\bar x}^ + }\bar x{f_x}\;\;\;{f_x}{\mathscr R}{{\bar x}^*} $

    可推断出$\overline {x{{\overline x }^ * }} = \overline x = \overline {{{\overline x }^ + }x} $.故$\left( {x{{\overline x }^ * }, {{\overline x }^ + }x} \right)$U

    $ \theta \left( {\left( {x{{\bar x}^ * },{{\bar x}^ + }x} \right)} \right) = {e_{x{{\bar x}^ * }}} \cdot {\bar x^ + }x = {e_x}{\bar x^ + }x = {e_x}x = x $

    这就说明θ是满射.所以θ是一个同构.

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    A Structure of Abundant Semigroups with Good Adequate Transversals
    KONG Xiang-jun1, WANG Pei2     
    1. School of Mathematical Sciences, Qufu Normal University, Qufu Shandong 273165, China;
    2. School of Software Engineering, Qufu Normal University, Qufu Shandong 273165, China
    Abstract: By means of an ${{\mathscr L}^ * }$-unipotent semigroup and an ${{\mathscr R}^ * }$-unipotent semigroup, a symmetrical spined product structure theorem for an abundant semigroup with a good adequate transversal is established. This conclusion removes "quasi-ideal"-an important premise-thus making it more concise than the conclusions in available literature. The result of this paper is an improvement and extension of the previous results about inverse transversals and adequate transversals. It lays a solid theoretical foundation for further study of the structure, properties and characterization of the congruences on this kind of semigroups.
    Key words: abundant semigroup    good adequate transversal    ${{\mathscr L}^ * }$-unipotent semigroup    
    X