cholrs Journl of Engineering nd Technology (JET) ch J Eng Tech, 015; 3(9):73-736 cholrs Acdemic nd cientific Pulisher (An Interntionl Pulisher for Acdemic nd cientific Resources) wwwsspulishercom IN 31-435X (Online) IN 347-953 (Print) Reserch Article *Corresponding uthor Yingqin Jin Rectngulr group congruences on n epigroup Yingqin Jin Deprtment of Computer nd Technology, Anhui University of cience nd Technology, Huinn, Anhui 3001, P R Chin Astrct: An epigroup is semigroup in which some power of ny element lies in su-group of the given semigroup The rectngulr group congruences on n epigroup re investigted A chrcteriztion of rectngulr group congruences on n epigroup in terms of its rectngulr group congruence pirs is given Moreover, it is proved tht the rectngulr group congruence on n epigroup is uniquely determined y its kernel nd hyper-trce Keywords: epigroups; Rectngulr group congruences; Kernels, Hyper-trces MR(000) uject Clssifiction : 0M10 INTRODUCTION Congruences on semigroups hve een the sujects of continued investigtion for mny yers In prticulr, congruences on regulr semigroup hve een explored extensively It is well known tht there re two principl pproches to study congruences on regulr semigroup: the kernel trce pproch nd the kernel norml system one Preston [1] introduced the kernel norml system for congruences nd chrcterized the congruences on inverse semigroups y mens of it Pstijn nd Petrich [] descried the congruences on regulr semigroups y their kernels nd trces In ddition, the concept of congruence pirs is nother effective tool for hnding congruences on regulr semigroups Gomes extended the kernel trce pproch to the kernel hyper-trce one nd used it to descrie the R- unipotent congruences nd the orthodox congruences on regulr semigroup in [3, 4] Moreover, Tn [5] descried the rectngulr group congruences on regulr semigroup y mens of the kernel hyper-trce pproch Using the wek inverses in semigroups, Luo [6, 7] generlized the corresponding results for regulr semigroups to eventully regulr semigroups The im of this pper is to estlish nlogues to the results in [5] We prove tht the rectngulr group congruence on n epigroup is uniquely determined y its kernel nd hyper-trce Furthermore, we give n strct chrcteriztion of rectngulr group congruences y mens of the rectngulr group congruence pirs PRELIMINARIE Throughout this pper, we follow the nottion nd conventions of Howie [8] Let e semigroup nd in As usul, E is the set of ll idempotents of, susemigroup of generted y E, E is the Re g is the set of ll regulr elements of nd V ( = { x, x xx} is the set of ll inverses of An element x of is clled wek inverse of if xx x Denote y W ( the set of ll wek inverses of in A semigroup is clled rectngulr group if it is regulr semigroup whose idempotents form rectngulr nd In fct, rectngulr groups re orthodox completely simple semigroups Let e congruence on semigroup The restriction of to the set E is clled the trce of denoted y tr Furthermore, the restriction of to the susemigroup E is clled the hyper-trce of denoted yhtr The suset { E( )} of is clled the kernel of denoted y ker Recll tht congruence on 73
Yingqin Jin, ch J Eng Tech, Decemer 015; 3(9):73-736 semigroup is sid to e regulr congruence if semigroup is sid to e rectngulr group congruence if is regulr semigroup In prticulr, congruence on is rectngulr group Let e semigroup nd e, f in E Define M( e, f ) { g E ge g fg} nd ( e, f ) { g E ge g fg, egf ef } ( e, f ) is clled the sndwich set of e nd f As is well known tht the set M ( e, f ) is nonempty for ll e, f E in n eventully regulr semigroup The set M ( e, f ) will ply n importnt role in eventully regulr semigroups s the set ( e, f ) in regulr ones The following concepts will ply fundmentl roles in this pper Definition 1 A susemigroup K of is sid to e norml if (1) K, K ; () E K ; (3), K, K K Definition A congruence on the susemigroup E of is sid to e norml if for ll x, y E, nd, then x y x y, x y whenever x ', y, x, y E Definition 3 Let e norml congruence on E such tht E is rectngulr nd nd K e norml susemigroup of Then pir (, K ) is sid to e rectngulr group congruence pir of, if it stisfies: (RCP1),, there exists ' such tht ', ' ' ; (RCP), x E, x K K Giving such pir (, K ), we define inry reltion (, K ) on y: (, ' (, K,, We denote y (, K ) { ' (, (, ' K,, RC () the set of ll rectngulr group congruences on nd denote y RCP () the set of ll rectngulr group congruence pirs of In the sequel, let is rectngulr group congruence pir of in order to simplify the nottion (, K ) The following lemms give some properties of n eventully regulr semigroup, which will e used in ection 3 ince the cse of epigroups is the specil cse of eventully regulr semigroups, the results on eventully regulr semigroups re lso true for epigroups Lemm 4 [6] Let e n eventully regulr semigroup If,,, ' nd g M ( ', ), then ' g Lemm 5 [9, 3] Let e n eventully regulr semigroup nd e congruence on (1) If ),,, then there exists such tht ; () If ef for some e, f E, then there exists g M( e, f ) such tht e gf ; (3) If is n idempotent of, then n idempotent e cn e found in such tht e Lemm 6 [10] A congruence on n eventully regulr semigroup is regulr if nd only if for ll exists such tht, where, E Lemm 7 [6] Let e n eventully regulr semigroup nd 3 Min Results We egin the section with the min result of this pper (, K) x, then W ( E Theorem 31 Let e n epigroup If ), then RC (, K ) ( ) congruence on such tht ker K nd htr (, K ) (, K ) E, there is the unique rectngulr group 733
Yingqin Jin, ch J Eng Tech, Decemer 015; 3(9):73-736 We follow the proof of Theorem 31 y series of lemms, where lwys represents n epigroup without extr illustrtion Lemm 3 Let (, K) ) nd, K If K, then x K for ll x E Proof Let K for some, Then K for ll By the condition (RCP), we get E Therefore K nd ( x K, ' x K for ny x E, ' It follows from the condition (RCP) tht x K Lemm 33 Let (, K) RCP ( ) Then is congruence on Proof We first show tht is n equivlence on It is ovious tht is symmetry The fct tht is reflexive follows from E K nd is reflexive To show tht is trnsitive, let nd c for some,, c Then for, there exists ' ' such tht ' K, ', ' ', nd for ', there exists c) such tht ' c K, c, ' c Therefore ' c, c since is trnsitive Put g M(, ) Then ' g K since K is norml nd K Moreover, we my otin g Es K, c K nd g E It follows from Lemm 3 tht ( g c K, so tht c K y (RCP) Dully, we my show tht for ll c), there exists such tht c ' K, c, c,nd so c, which implies tht is trnsitive Consequently, is n equivlence on We now show tht is congruence on To show tht is left comptile, Let with some, For ny ( c, then ' ( c, c), (, c It follows from tht ( c K And since there exists ' such tht y the definition of, it follows from Lemm 5 tht there exists g M(, ) M( c, ) such tht c nd g Put ( g Then ( g c, which gives tht ( c( c c( g) nd c( ( ( c g g ( g c ( ( c A similr rgument will show tht for ny ( c, there exists ( ' c such tht ( c K, c ' c(, ( c )'' c ( c Hence c c, together with the fct tht is n equivlence, nd so is left congruenceon On the other hnd, the ssertion tht is right congruence on cn e shown dully Thus is congruence on, s required We now estlish the connection etween rectngulr group congruence nd its kernel nd hyper-trce Lemm 34 Let (, K) ) Proof To prove tht Then htr nd is rectngulr group congruence on htr, let xhtry for some y E x, ince is rectngulr nd congruence on E, we, y Lemm 7, conclude tht there exist E, y' y) E such tht ' x x x, xx yy', y 734
Yingqin Jin, ch J Eng Tech, Decemer 015; 3(9):73-736 Hence x x x yy' x yx nd y yy' y ( since is rectngulr nd congruence on E ) y x yx, nd so x y, which leds to htr Conversely, let x y for some x, y E Assume tht E Then x ' y E K, nd so y x from the fct tht is rectngulr nd congruence on E Hence y ),nd so there exists y' y) E such tht y' y Lemm 5 It follows tht xx ' yy', y Furthermore, for,there exists such tht y ' x K, x yy', y,nd so x y, which leds to htr Thus we deduce tht htr, s required We now show tht (, K ) is regulr congruence on For,, there exists ' such tht ' ' K, ' so tht ( ( ' ) It follows from Lemms 5 nd 7 tht there exists ( ' ' ' E such tht ( '' ' And since ( ' '( ' ( ' ' ( ' '( ' ' ' ' ' ( ' '( ' ' ( ' ' ( ince E is rectngulr nd nd htr ),which implies tht (( ' ' ) ( ' ) It follows from Lemm 5 tht there exists ( ' ' ' such tht ( ' ' ( ' ', nd so ' ' ' ' ' ' ' ', ( 由 RCP1) which leds to ' ' ' ' y htr Furthermore, we otin ( ' ' ' ( ' ' ' ' ', ( 由 RCP1) nd so ( '' ' ' ' y htr For c ', then c ', c W ( '') E K, c ' ' c c ', nd so ( c '' ( c ) Hence there exists x c E such tht x c ' Put xc Then, so tht ( xc) ' xc '' c ( ' c nd ( xc) c( ' Hence ', nd so it follows from Lemm 6 tht is regulr congruence on such tht To show tht is rectngulr group congruence on, let, E( ) Then there exist e, f E e, ef, nd so efee from the fct tht E is rectngulr nd Thus (, nd so is rectngulr group, which gives tht is rectngulr group congruence on Lemm 35 Let (, K) ) Then ker K Proof To prove ker K, let ker for some e E such tht e Then there exists exists e' e) E such tht e' K Hence K y (RCP), so tht ker K Conversely, if since K is norml susemigroup K, then K, nd so there 735
Yingqin Jin, ch J Eng Tech, Decemer 015; 3(9):73-736 of, nd so K uppose e M(, ) It follows from Lemm 4 tht e ) And since is regulr congruence on, together with Lemm 7, there exists ' such tht ' Hence where where ( e) e ' ', e,, ' E ( e) '( e) ' e, E (ince is rectngulr group congruence), so tht ' e y htr Notice tht (ince is rectngulr group congruence), so tht ( e) y htr On the other hnd, let c, ' ' c for ny c ) Then, '' nd c '', ' ', nd so c '' K since K nd K is norml susemigroup of Hence we,since E is rectngulr nd, otin '' '' ' ' nd '' ) ( ', nd so '' '' ' ' ) ( 1) It follows from tht ( ( ), nd so there exists x E such tht x Notice tht ( c cc) c x cx, so tht cx It follows tht c ( ' ' ' '' c ( y() 1 ) In similr wy, we get ( c '' c Therefore K ker, s required Up to now, Theorem 31 is direct consequence of Lemms 33, 34 nd 35, nd so K ker, which implies tht ACKNOWLEDGEMENT: This reserch is supported y the Tlent Youth s Foundtion of Eduction Deprtment of Anhui Province (011QRL040) REFERENCE 1 Preston GB; Inverse semigroups J London Mth oc, 1954; 9: 396-403 Pstijn F, Petrich M; Congruences on regulr semigroups Trns Am Mth oc, 1986; 95: 607-633 3 Gomes GM; R-uniotent congruences on regulr semigroups emigroup Forum, 1985; 31: 65-80 4 Gomes GM; Orthodox congruences on regulr semigroup emigroup Forum, 1988; 37: 149-166 5 Tn X; Rectngulr group congruences on regulr semigroups Pure nd Applied Mthemtics, 00; 18:161-164 6 Luo YF; R-uniotent congruences on n eventully regulr semigroup Alger colloquium, 007; 14(1): 37-5 7 LuoYF; Orthodox congruences on n eventully regulr semigroup emigroup Forum, 007; 74(): 319-336 8 Howie JM; Fundmentls of emigrops Theory, Oxford: Clrendon Press, 1995 9 Auinger K, Hll TE; Representtions of semigroups y trnsformtions nd the congruence lttice of n eventully regulr semigroup, Int J of Alger nd Computtion, 1996; 6(6): 665-685 10 Weipoltshmmer B; Certin congruences on E-inverse E-semigroups emigroup Forum, 00; 65: 33-48 736