IRJET- On Semigroup and its Connections with Lattices

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ON SEMIGROUP AND ITS CONNECTIONS WITH LATTICES Dr. Pankaj Kumar Chaudhary1, Dr. Jawahar Lal Chaudhary 2 , Gyan Shekhar 3 1Assistant

Professor, Department of Mathematics, Women’s Institute of Technology, L. N. Mithila University, Darbhanga, Bihar, India 2 Associate Professor, University Department of Mathematics, L. N. Mithila University, Darbhanga, Bihar, India 3 Research Scholar, Department of Mathematics, L. N. Mithila University, Darbhanga, Bihar, India - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - * * *- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ABS TR AC T : - L. V. Sh i vr i n a n d B. M. Ve r n ik o v d er iv e d pr ope rt ies of s e m i gro u p va r iet ie s w h ic h f or ms a la t t ic e. W e cr it ic a l ly e x a m i ne t he d if fe re nt c la ssi f ica ti o ns o f mo d u lar va r iet i es. I t is k n o w n that th e co ll ect i on S EM o f a ll s e m i gr ou p va r iet ie s f or m s a latt ic e wi th r esp ect to c lass th eo ret i cal i n c l us i o n. A sem i g ro up var i et y mo d ul a r is ca l le d lo w er - m o d ular, d i str i b ut iv e if it is a m o d ul ar l o we r - mo d u lar, d ist r i b ut i ve e le m e nt o f t h e la t t i ce SE M. D i str i b ut iv e va r iet ie s ha ve b ee n dete r m i ne d L. N. S h erv i n we d is cu ss p rop ert i es of a cla ss of l o we r - m o d ul a r va r iet ie s. B. V. Ve r n i ko v d er i ve d it s pr op er tie s, we exa m i n e a c o mp let e cla ssi f ica t i o n o f l o wer - m o d ul ar va r iet ie s. T he ma i n res u lt o f th is art i cl e gi v es a co mp let e clas s ifica t i o n of l o we r -m o d ul a r va ri e tie s. I f B is a s et o f i d e nt it ie s, t he n ∼ B de n ote s th e f ul l y i n va r ia nt c o n gr ue n ce o n t h e f ree s e m ig ro up co rresp o n d i n g t o B. we es tab li sh t he c o n ne ct io ns bet w ee n t he qu as i -or d ers ≤ B a n d ≤.

K EY WO R DS : S em ig r ou p, L a t ti c es, S e m il a t tic e , M odul a r , Q ua si- o rd e rs IN TR O D UC T IO N An el em e n t x o f a l a tt ic e  L ,  ,  is c a l l ed mo d ul a r i f (

)

(

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... (i)

l ow e r- mo dul a r i f (

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(

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(

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…( ii)

dis t ri bu ti v e if (

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(

) …( ii i)

Up p er- m od ul a r el em e n ts ma y b e d e fi n ed d u a l l y t o l o w er - m odul a r o n es. H e nc e , w e fi n d tha t a dis t ri bu ti v e el em e n t is l o w e r - mo dul a r . A pa i r of id e nt it i es w x = xw = w is k no w n a s a n i d e nt it y w = 0. S i nc e j ust ifi e d b ec a u se a s e mi gr ou p wi th s u ch id e n ti ti es ha s a z e r o el e m e nt a n d a l l va l u es of th e wo r d w in thi s s e m ig ro up a r e e qu a l to z e r o. I d e nt it i es o f th e fo r m w = 0 m or e ov e r a s va ri e ti es g iv e n by such i d e nt it i es a r e c a l l e d 0 - r e duc e d. By T, SL , a nd SE M w e d en o t e th e t ri via l va r i et y, th e va r i et y of a l l s e m il a t tic es , a n d th e va ri e ty o f a l l s e mi gr ou ps , r es p ect iv el y . W e p r ov e h e r e th e f ol l o wi n g th eo r e m. L et us r ep r e se n t by F  th e f r e e s e mi gr ou p a b ov e a cou n ta bl y i n fi n it e a l pha b e t, i. e. th e se mi g ro up o f wo r ds u n d er c o nc a t e na t io n. I f B is a s e t of r e gul a r id e nt it i es a n d v a nd u a r e w or ds , l et us d e fi n e v ≤ B u if a nd o nl y i f va r { v ≈ 0, B } |= u ≈ 0. If t h e s e t B con ta in s o nl y tr iv ia l i d e nt it i es, th e n i ns t ea d o f ≤B L e t us w ri t e ≤ . Th e r el a ti on ‘ ≤ ’ o n th e f r e e s e m ig ro up is k n o w n a nd ma y b e d ef in e d a s f ol l o ws: i f u , v ∈ F ∞ , th e n v ≤ u i f a nd o nl y if u = aΘ( v) b f o r s om e p ossi bl y empty words a a nd b a n d s om e su bs ti t uti o n Θ w e fi n d t ha t th e r el a t io n ‘ ≤’ B is r efl e xiv e a n d tra ns i tiv e , i . e. it is a qua s io r d e r o n th e fr e e se mi g ro up F ∞ . I f u ≤ B v ≤ B u , th e n u⇔ B v . If u is a wo r d, th e n th e cl a ss of a ll wo r ds e q uiva l e n t to u m od ul o ⇔ B is d e no t e d by [ u ]⇔ B . L et F ∞ / ⇔ B d e n ot e t h e s e t of a l l cl a ss es [u ] ⇔ B o r d er e d by ≤ B . t he el e m en ts o f th e o rd e r ed s e t F ∞ / ⇔ B i s ca l l e d w or d patt er ns mo d u lo B. Th e o re m 1 I f u ≤ v, t he n | u | − |Co n t ( u) | ≤ | v | − |C o nt (v ) | .

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Pr oof Ass er ti o n o f th e l em ma s is v e r if i ed by ta k i ng t he fol l ow i ng tw o ca s es.

Ca se i: v = u x o r v = x u fo r so m e n ew va r ia bl e x.

Ca se ii: v = Θ( u) wh e r e Θ is a n el em e n ta ry su b sti tu ti on .

Al so, th e g e n e ra l c a s e f ol l o ws by t ra ns it ivi ty o f r el a ti on ‘≤ ’ o n wo r d p a tt e r n.

Th e o re m 2 I f B is a s e t of r eg ul a r i d en ti ti e s , t h en th e f ol l ow i ng s ta t e m e nts a r e e qu iva l e nt : (i) v ≤ B u ; (ii) t her e e x ists a w or d u  s uc h t ha t u ∼B u  a n d v ≤ u .

W e f i rst pr ov e im pl ic a t io n (i) ⟹ (i i).

Pr oof (i) → ( ii) , I f v ≤B u, th e n t h e r e ex ists a s e qu e nc e of w or ds u 1 , . . . , u n s uch tha t u ≈ 0 f ro m wh ic h i t fol l ow s tha t u = u 1 ≈ u 2 ≈ · · · ≈ u n ≈ 0. Si nc e m (B) > 0 bu t m( u ≈ 0) = 0 , u ≈ d er iva bl e f ro m B w ith o ut us i ng v ≈ 0. By us i n g th e i d en ti ty v ≈ 0 t h e d er iva ti o n o f u ≈ 0 is So , u ∼B u n a n d v ≤ u n . (ii) → (i) . fol l o ws a s a n im m ed ia t e c o ns e qu e nc e o f c o nd it io n ( i). I f v ≤ B u but u ∼ B v , th e n v < B u.

{ v ≈ 0,B} 0 i s no t possibl e. we write

Th e o re m 3 I f B is a s et o f ba la nc e d i de nt it ie s , t he n t he f ol l o wi n g c o n d it io ns a re sa tis f ie d. (i) (ii) (ii i) (iv)

u ⇔ B v i f an d o nl y i f u ∼ B p (v ) f or so m e re na m i n g p o f va ri ab le s ; i f u ⇔ B v, th e n | u | = | v |; i f v ≤ B u a n d | v | < | u |, t he n v < B u ; v < B u i f a n d o n ly i f t he re e xi st s a w or d u  s uc h tha t u ∼B u a n d v < u 

Pr oof W e p r ov e ea ch s ta t e m e nt s e pa ra t el y. (i) I f v ∼B p( u) , t h en u ≈ p( v) ≈ 0 is a d e r iva t io n of u ≈ 0 f ro m { v ≈ 0, B} . Th e i d e nt it y v ≈ 0 ca n b e d er iv e d f ro m { u ≈ 0, B } i n a s im il a r wa y . S o, u ⇔ B v . If v ⇔B u, t h en us in g a ss e rt io n , l e t us o b ta i n wo r ds u  a n d v suc h t ha t u ∼ B u , v ≤ u a n d v ∼ B v , u ≤ v . Si nc e th e id e n ti ti es u ≈ u a n d v ≈ v a re ba l a nc ed , a l l th e wo r ds u, u , v a n d v ha ve th e sa m e l e n gt h. Th e r ef o r e, u = p( v) f or so m e sub st it ut io n p tha t ma ps va ria bl es t o va ria bl es . L e t us su p pos e tha t fo r s om e dis ti nc t va r ia bl e s x a nd y w ith oc c v (x ) > 0 a nd o c c v (y ) > 0 w e ha v e tha t p ( x) = p( y) = z . But th e n | u | − |C o nt (u ) | = |u| − |Co n t ( u ) | > |v | − |C o nt (v) | = | v | − |C o nt (v  ) |. B y a n a p pr o pr ia t e a p p l ica t io n o f L e mma ( 4. 3. 5), t his c on t ra d ic ts th e fa c t t ha t u ≤ v . T h e r ef or e , th e sub st it ut io n p is a r e na mi n g of va r ia bl es. (ii) I f u⇔ B v, th e n b y u sin g pa r t (i) t h e id e n ti t y u ≈ p( v) i s ba l a nc e d. Th e r ef o r e, | u | = |p( v) | = | v| . (ii i) If u ⇔ B v , th e n by p a rt ( ii) w e o b ta i n t ha t | u | = | v |. T h e r ef or e v < B u. (iv) Si nc e v < B u , By su ita bl e a p pl ic a t io n o f l e mma ( 4. 3. 5) w e ma y f i nd a wo r d u s uch t ha t u ∼B u a nd v ≤ u . S o, u = a Θ ( v) b f or s om e po ssi bl y e mp ty wo r d ab a n d a s ubs ti tu ti o n Θ . I f th e w or d a b is e mp ty a n d Θ i s a r e na mi n g o f va r ia bl es, th e n , by usi n g pa r t ( i), w e f in d tha t u⇔ B v. Si nc e t he w o rds v a n d u a r e n o t e qu iva l e nt mo dul o ⇔B , it im pl ie s v < u .

L e t us s up po s e tha t th e r e is a w o rd u s uc h t h a t u ∼B u a nd v < u . T he n by a n a p pl ica tion of L emma (4. 3.5 ), w e g e t v ≤ B u . I f |v | < | u | th e n by su ita bl e a p pl ica ti o n of pa rt (ii i), w e fi n d tha t v < B u. L et us a ssum e tha t | u | = | v | a nd c o ns e qu e n tl y | u| = |v | = | u | . S in ce v <u a nd | v | = | u | w e ha v e tha t u = Θ (v) fo r so m e su bs ti tu ti on Θ t ha t ma p s va r i a bl es t o va ria bl es su c h tha t Θ i s n ot a r e na mi n g of va r ia bl es. So , u ∼ B Θ (v ) . L et us fu rt h er a ssu m e tha t u ⇔ B v th e n by p a rt (i) tha t u ∼ B p (v ) f or so m e

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r e na m i ng of va r ia bl es p . Th us t h e i d en ti ty Θ( v) ≈ p( v) a l s o ba l a nc e d w hich is no t t he ca s e .

Th e r ef o r e, v < B u . H e nc e, th e l em ma is es ta bl is he d.

Th e o re m 4 I f B is a s et o f ba la nc e d i de nt it ie s w it h ℓ ( B) > 0 , t he n: (i) the wo r ds x 2 y 1 · · · y k a n d y 1 · · · y k x 2 are n ot e q u iva le nt m o d ul o ⇔ B f o r a n y k > 0; (ii) the wo r ds x k y a n d y x k a re n ot e q u iv al en t m o d ul o ⇔ B fo r a ny k > 0; (ii i) t he wo r ds y 1 · · · y j x k + 2 t 1 · · · t p a n d y 1 · · · y j + 1 x k + 1 t 1 · · · t p are n ot equivalent modulo ⇔ B for any k ≥ 0.

Pr oof (i) If x 2 y 1 · · · y k ⇔ B y 1 · · · y k x 2 , t h e n L e m ma (4. 3.5 ) i mpl i es x 2 y 1 · · · y k ∼ B p (y 1 · · · y k x 2 ) f or s o m e r e na m i ng o f va ria bl es p. Si nc e x is t h e o nl y n on - l i n ea r v a r ia bl e i n b oth wo r ds, thi s ca n o nl y ha p p en if B ⊢ x 2 y 1 · · · y k ≈ y 1 · · · y k x 2 whi c h is i m pos s ibl e in v i e w o f L e m ma ( 4.3 .5) . (ii) Si mil a r t o th e p ro o f of ( i). (ii i) If y 1 · · · y j x k + 2 t 1 · · · t p ⇔ B y 1 · · · y j + 1 x k + 1 t 1 · · · t p t h en by L em m a (4 .3. 5) we ha v e tha t y 1 · · · y j x k + 2 t 1 · · · t p ∼ B q( y 1 · · · y j + 1 x k + 1 t 1 · · · t p ) fo r som e r e na mi ng o f va ri a bl es q. Si nc e x is th e onl y nonl in ea r va r ia bl e i n b ot h w or ds a nd oc c u rs d i ff e r e nt nu m b e r o f ti m es i n ea ch of th e wo r d, t his is im po ssi bl e i n v i ew of L e mma ( 4. 3.6 ).

Th e o re m 5 If L i s a l a tt ic e a n d a  L t h e n [a ) s ta nds fo r t he p r i nci pa l c oi d ea l g e n e ra t e d by a , t ha t is , t he s et { x  L | x  a }. If x i s a l ow e r - mo dul a r el e m e nt of a l a tt ic e L a nd a  L th e n t h e el em e n t x  a is a l ow e r- mo dul a r el em e n t o f th e l a tt ic e [a ).

Pr o of L e t y, z  [a ) a n d x  a  y . Th e n , w e ha v e (

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(

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( ( ( ( ((

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Thu s ( x  a )  ( y  z) = y  (( x  a )  z) , H e nc e, th e th e o r em is p r ov e d .

REFERENCES [1] V. Yu. Shaprynskiı and B.M.Vernikov, Lower-modular elements of the lattice of semi--group varieties. III, Acta. Sci. Math. (Szeged), accepted (1996) [2 ] M. V.Volkov, Modular elements of the lattice of semigroup varieties, Contrib. General Algebra, 16, pp275–288 (2005) [3 ] Vernikov B.M., Volkov M.V., Modular elements of the lattice of semigroup varieties II, Contributions to General Algebra, 17, pp. 173–190, Heyn, Klagenfurt (2006) [4 ] C. Herrmann and M.V. Semenova, Existence varieties of regular rings and complemented modular lattices, J. Algebra 314, no. 1, 235–251(2007) [5 ] B.M.Vernikov and V.Yu. Shaprynskiˇı, Distributive elements of the lattice of semigroup varieties, Algebra i Logika, 49, 303–330, in Russian; Engl. translation: Algebraand Logic, 49, 201–22(2010) [6 ] Vernikov, B.:, Proofs of definability of some varieties and sets of varieties of semigroups. Semigroup Forum 84, 374– 392 (2012) [7 ] Grech, M., The structure and definability in the lattice of equational theories of strongly permutative semigroups.

Trans. Am. Math. Soc. 364, 2959–2985 (2012)

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