Semilattices of Rectangular Bands and Groups of Order Two.

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1 1 Semlattces of Rectangular Bs Groups of Order Two R A R Monzo Abstract We prove that a semgroup S s a semlattce of rectangular bs groups of order two f only f t satsfes the dentty y y, y y, y S 1 Introducton AMS Mathematcs Subject Classfcaton (010): 18B40 Key words phrases: Semlattce of semgroups, Incluson class, Rees matr semgroup Semgroup theory has a certan symmetrc elegance that lnks t to group theory For eample, a semgroup S s a group f ( only f) for every S, S S S In addton, a semgroup S s a unon of groups f ( only f) for every S, S S[] The powerful result that a semgroup s a unon of groups f only f t s a semlattce of completely smple semgroups s so well known that ts beauty can almost be overlooked [] In ths paper we apply ths result to prove that the collecton of all semgroups that are semlattces of semgroups that are ether rectangular bs or groups of order two s a semgroup ncluson class [4] Precsely, a semgroup s a semlattce of rectangular bs groups of order two f only f t satsfes the dentty y y, y y, y S Notaton, defntons prelmnary results Defnton Let G be a group LetS I G I a non-empty set Let : I G P, wth P, j defne a product on S as follows:, a, j, b,, ap jb, denoted by p j Then S endowed wth ths product s called a Rees I matr semgroup (over the group G wth the swch matr P) The well known defnton above s repeated here because t wll be used etensvely throughout ths paper All other termnology notaton can be found n [] Other well known results that wll be used here follow Result 1 [5, Corollary IV8] A semgroup s completely smple f only f t s somorphc to a Rees matr semgroup Result [,Theorem 4] A semgroup S s a unon of groups f only f for every S, S S Result [, Theorem 46] A semgroup s a unon of groups f only f t s a semlattce Y of completely S I G Y smple semgroups Result 4 [, Proposton 1] A semlattce Y of completely smple semgroups s a semlattce of rectangular groups f only f the product of two dempotents s an dempotent Defnton We wll denote the collecton of all left zero, rght zero rectangular bs by L 0, R 0 RB respectvely that of all groups of order two by G If S s a unon of groups S then 1 wll denote the dentty element of any group to whch belongs If G s a group then the dentty element of G s

2 1 denoted by 1 or 1 G Also f s an element of a semgroup S then 1 or denotes the nverse of n G any subgroup G of S A rectangular group s the drect product of a rectangular b a group Note that f an element of a semgroup belongs to two groups, G H, then , therefore1 s well-defned G H G H G G G H H H Smlarly, 1G 1H 1G 1H 1 s well-defned so G G G H G H H H Defnton [4] An ncluson class of semgroups s a collecton of all semgroups that satsfy a gven set of k number of nclusons as follows: (1),,, n,,, 1 m1 (),,, n,,, 1 m (k),1,,,,,,1,,,,, W w w w t t t T ; 1 1,1 1, 1, 1,1 1, 1, 1 W w w w t t t T ; ;,1,,,1,, W w w w t t t T, where the w s the t s are semgroup words over k k k k n1 k k k mk k some alphabet Notaton We wrte [ W1 T1 ; W T ;; Wk Tk ] to denote the ncluson class determned by the set W T 1,,, k, S then we wrte f of nclusons If S s a semlattce < f Defnton A semgroup S s a semlattce f S[ ; y y ] A semlattce S s a chan f, S mples, Defnton A semgroup S s a semlattce Y of semgroups S Y S Y, Y,S S S for every f S s a dsjont unon of the Some ncluson classes of semlattces of rectangular bs groups of order two Theorem 1 The followng statements are equvalent: 1 S[ y, y ] S s a chan Y of semgroups S Y (1) S RB G Y, () for any <, Y () for any <, Y where any S, y S, y y wth S G, S 1

3 Proof: 1 Let S[ y, y ] Then clearly, S[ therefore, by Result, S s a semlattce Y of completely smple semgroupss Y Let S y S, Y ] so by Result, S s a unon of groups, Then ether (a) y yy y or (b) y yy or (c) y y yy y or (d) y y yy In cases (a) (d), So f then ether case (b) or (c) holds Therefore, mples that ether or Hence,Y s a chan We now show that S RB G Y Each S I G Y Then let, g, y j, h,, where ether j or G Now (b), (c) (d) all mply that j Thus, (a) holds Now so 1 Assume that ether I 1 or 1 where g h are arbtrary elements of 1 So, g,, gp g,, p, Hence, gp g p 1 Let g be an arbtrary element of G Snce g was arbtrary we can let g g p 1 Then p g p p g p hence g s the dentty element of G therefore G G Hence, G s abelan p p Now (a) holds so y So, g,, gphpg, therefore g gphpg 1 p j hp g Settng g 1G h gves G 1 p p Therefore G 1 Ths mpless RB Now f But G s abelan so then the mappng I As shown two paragraphs above, G G so S G so (1) s vald Now assume that, Y 1 1 Then 1 yy 1 1 If S G y g gp hp g g p p h h 1 g, g p, s an somorphsm between G S We have therefore shown that S RB G Y let S y S Then (b) holds so y yy thens s commutatve so yy 1y1 11 y 1 y z 1 z z yz 1 Therefore S y y y y y y y y y y y y However, f z S vald) If S z RB then1 y yy y y then so 1 y y (We have shown that () s Also, from (b), y y yy y y Hence, y y We have shown that () s vald ths completes the proof of (1 ) 1 Assume thats s a chan Y of semgroups S Y We wsh to show that y, y for any, y S SnceS RB G Y where (1), () () n Theorem 1 are vald, we can assume that

4 4 Case 1 Suppose that < S G Case Suppose that < S Case If < S Case 4 As n the proof of case, < S We have therefore shown that y, y Then by (), S so y S RB Then G then, by (), S y Therefore, ys y y yy y y y y y y Theorem 1 Note that we have shown that anys[ y, y y y y y RB mples yy y Then, completng the proof of ( 1) Ths completes the proof of ] s a chan Y of rectangular bs, ecept possbly f Y has a mamal element S G wth S >1 The queston arses as to whether the collecton of chans of semgroups that are ether rectangular bs or groups of order s an ncluson class In Theorem 5 below we prove ths queston n the affrmatve when the word chan s replaced by semlattce Theorem The followng statements are equvalent: 1 S[ y y, y ] S s a semlattce Y of semgroups S Y (1) S R G Y, 0 where () < S G mples S 1 () S S mpless 1,, (4) for each Y S there s a mappng :S S such that for any Y, S / :S S s a homomorphsm, satsfyng (41) f S G then for any g S y, S, g 1 g y on S, (4) f,, Y S R0 then y y y on S (4) for every S y S, y y y Proof: 1 Assume that S[ y y, y ] Frst we wll prove thats[ y y ] Let, y S Then ether (a) y y yy or (b) y y yy y or (c) y y yy or (d) y y yy y Note that snces[ y y, y ], y y, y So n each case we can assume that y y Case (a): y yy y y y yy y y y y 9 6 4

5 5 Case (b): y y y y yy y Case (c): y y y yy yy But then y mples y y y y Case (d): y yy y y y y y y By Results, S s a semlattce Y of completely smple semgroups S Y product of two dempotents of S s an dempotent Let e, f ES Then efe f, fe ef eefe efe f ES If efe fe thenef efe f fef e, ef Hence ef ef f ef Now by Result 4, y so We now show that the If efe f then So we can assume thatef e ths completes the proof that the product of two dempotents s dempotent S s a semlattce Y of rectangular groupss Y S LG R s ether a rght-zero semgroup or a group of order two Let, y S wth, g, y j, h, Then y y, y We have shown that an arbtrary rectangular group component of We want to show now that each so j So S s a rght group G R S G R Note that, snce we have already shown above that y y satsfes g g 1, g 1 ghg h g, h G, G G so G s commutatve Now let g g, r g, s G R, wth r s g 1 So ether R 1 For Y we defnes S : S y Then g, r g, r g, s g, r g, s, g, r Therefore or G 1 Hence G R s ether a rght-zero semgroup or a group of order Let, y S wth S y S We show that S S Frst note that S s a null etenson of a unon of groups From the proof of Theorem 5 [1], y y S S y It s then straghtforward to show that (1) y y1 y1 y 1 y1 y 1 1 [ ], wth1 y y S 1 S y Assume that y S that Then, snce S s a semlattce of the semgroups S Y, t follows from (1) Note that S y S Therefore ThereforeS s a semlattce of the semgroupss Y we have proved part (1) of Theorem y y y, whch mples that It s straghtforward to show thats S Thus,

6 6 Suppose now that, S S S y G so 1 G y S Suppose that S S Let S of Theorem y Then y y, y ThereforeS 1 We proceed wth the proof of part (4) For any / S :S S, because y y y y S S R 0 then so y y But Ths proves part () of Theorem Then by hypothess y y, so 1 Ths proves part () S we defne :S S S S as y y y S Lety, z S If Note that yz y z y z yz z z y z z y z y z y z Suppose that S G that If yz, y, z S 1 then ether < or < so, by (), S 1 so / S s a homomorphsm so Ths mples We can therefore assume that S S S G Snce, by 4 hypothess,,, 1 Then, y z y z yz yz so / S s a homomorphsm n any case Ths proves (4) LetS G, g S,1 1 g, y, z S 4 4 y z yzy y yz y yz y yz y z y z 1z 1z1 1 z Theng gg g11g g Also, 4 z z 1z 1z1 1 1z 1 1z 1z z z z z 1z1 1z Hence, ths proves (41) Let g 1 g y,, Y S R0 Suppose S, y S z S Then z z y z y yzy yzy yzyy zy zy yzy zy y y y yy yz yzy yzy z Ths proves (4) y Fnally, (4) follows from the fact that y y Ths completes the proof of1 1 Assume that the hypotheses of the only f part of Theorem are vald We frst show that the product defned s assocatve Let S, y S, z S Usng (4) we need to show that: z y y z z y z y y y y z z y However, snce by (4) z are homomorphsms on S S respectvely, ths equaton becomes: () z z yz y yz z y, y y y z z y

7 7 wth by (4) agan -- each of the 6 terms an element of (4 ), z y z y y z, equaton () s vald n ths case S IfS R0 then, snce by hypothess So we can assume thats G We can therefore assume that, or else there ests,, such that <, whch mples S 1 [Ths would mply that equaton vald] But f then, by (41 ), each sde of equaton equals, z y, so Now we need to prove that for S y S, y y, y y y y y y We can assume, therefore, thats G s s vald If S R0 then, sncey, y S, If then ether < or < In ether casey, y S 1 so y y We can assume therefore that We can also assume that y S y y S G, snce y Also, 1 Hence, y y y1 y So the proof of Theorem s complete Corollary The followng statements are equvalent: 1 S s a semlattcey of semgroupss Y (11) S R G Y, 0 (1), Y, S G, or else by (),y, y S 1 Note that S s an abelan group Therefore, y y y y y y Ths completes the proof thats[ y y, y where mples S 1 (1) there s a collecton of mappngs :S S / S (a) for, Y each / S :S S s a homomorphsm, ] satsfyng the followng propertes: (b),, Y, S, y S S R mply 0 y y y on S y (c) fs G y y, y ; ] S[, S then 1 y on S

8 8 Theorem 4 The followng statements are equvalent: 1 S s a semlattcey of semgroupss Y S[ y y y y, ; ] wths R 0 G Proof:1 For any R0 G, therefores[ ] Suppose that S y S then y y y y y If S R0 Suppose thats G Then, y y yy y y yy y y y 1 y yy 1 y yy 1 y 1 y y y1 y1y y y Also, y y y yy y y Then, 1 1 We have proved therefore that S[ y y, y y ] [ y y, y y ] y y y y y y y S[ 1 Ife, f ES, Y ] Hence, then, snce efe fe, fef, ef ef ef efe f ef fe f, ef fef f ef It then follows from Results 1, thats s a semlattcey of rectangular groupss Y LetS L G R y y, y y ] mples L 1 so The fact thats[ ], G [ ] G G,,,,,,,, S G R SnceS[ g, r, y h, s S Then Let y ghg r h r hg r h g h s hg r h s So R 1mples G 1 Thus, ether R 1or G 1 SoS R0 G, whch s what we needed to prove Ths completes the proof of Theorem 4 Theorem 5: The followng statements are equvalent: 1 S s a semlattce Y of semgroups S Y wth S RB G Y S[ y y y y, ; ] Proof: 1 Clearly, for any S Let, Y wth S y S IfS RB then y y y y y yy y So we can assume thats G y 1 y1y 1y y 1y y 1 y 1 Then y y 1 1y 1 y 1 y 1 y Therefore, 1 y y 4 Then, y y y y y y yy yy y y y y y y y y y 1 y1 1 y 1 y y y y y1 1y1 1y y y Hence, S[ yy, y y ; ] 1 SnceS[ ], by Results, S s a semlattce of completely smple semgroups

9 9 We now show that the product ef of the dempotents e f s dempotent We have,,, 5 6 efefe efe fef efe efe fef efe fef efe fef ef e fe f efe fe f If efefe efe thenef efe f efefe f ef ef So we can assume that Then, efefe fe f fe f e fe fe fe f f efefe f ef ef Therefore,ef e fe f eef f ef groupss G E Y,where E RB Y efefe fe f So by Result, S s a semlattce Y of rectangular F Y let, y S wth 1,, y h, j, of G, j, are arbtrary elements of E Then, where h s an arbtrary element y h,, y, y y h,,, h, j, So ether E has only one element or h h for every h G HenceS s somorphc to G or to E In the former case, snce S[ ], G G Ths completes the proof of Theorem 5 References [1] Clarke, GT, Monzo, RAR,: A Generalsaton of the Concept of an Inflaton of a Semgroup Semgroup Forum 60, (000) [] Clfford, AH: The Structure of Orthodo Unons of Groups Semgroup Forum, 8-7 (1971) [] Clfford, AH, Preston, GB: The Algebrac Theory of Semgroups Math Surveys of the Amercan Math Soc, vol 1 Am Math Soc, Provdence (1961) [4] Monzo, RAR: Further results n the theory of generalsed nflatons of semgroups Semgroup Forum (008) [5] Petrch, M: Introducton to Semgroups Charles E Merrll Publshng Company Columbus, Oho (197)