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UNIVERSIDADE ABERTA Conjugation in Abstract Semigroups UNIVERSIDADE ABERTA Conjugation in Abstract Semigroups Maria de F´atimaLopes Borralho Doctor's Degree in Computational Algebra Doctoral dissertation supervised by: Professor Jo~aoJorge Ribeiro Soares Gon¸calves de Ara´ujo Professor Michael K. Kinyon 2019 ii Resumo Dado um semigrupo S e um elemento fixo c 2 S, podemos definir uma nova opera¸c~aoassociativa ·c em S por x ·c y = xcy para todo x; y 2 S, obtendo-se assim um novo semigrupo, o variante de S (em c). Os elementos a; b 2 S dizem-se primariamente conjugados ou apenas p-conjugados, se existirem x; y 2 S1 tais que a = xy e b = yx. Em grupos, a rela¸c~ao ∼p coincide com a conjuga¸c~aousual, mas em semigrupos, em geral, n~ao´etransitiva. Localizar classes de semigrupos nas quais a conjuga¸c~ao prim´aria´etransitiva ´eum problema em aberto. Kudryavtseva (2006) provou que a transitividade ´ev´alidapara semigru- pos completamente regulares e, mais recentemente, Ara´ujoet al. (2017) provaram que a transitividade tamb´em se aplica aos variantes de semigru- pos completamente regulares. Fizeram-no introduzindo uma variedade W de epigrupos contendo todos os semigrupos completamente regulares e seus vari- antes, e provaram que a conjuga¸c~aoprim´aria´etransitiva em W. Colocaram o seguinte problema: a conjuga¸c~aoprim´aria ´etransitiva nos variantes dos semi- grupos em W? Nesta tese, respondemos a isso afirmativamente como parte de uma abordagem mais geral do estudo de variedades de epigrupos e seus variantes, e mostramos que para semigrupos satisfazendo xy 2 fyx; (xy)ng para algum n > 1, a conjuga¸c~aoprim´ariatamb´em´etransitiva. Numa fase inicial da tese foi feita uma revis~aoda literatura referente `a problem´aticaem estudo com uma breve discuss~aoe declara¸c~aodos princi- pais resultados. No Cap´ıtulo2, referimos que seguindo Petrich e Reilly (1999) para semigrupos completamente regulares e Shevrin (1994, 2005) para epi- grupos, ´ehabitual ver um epigrupo (S; ·) como um semigrupo un´ario(S; ·; 0) em que x 7! x0 ´ea aplica¸c~aoque a cada elemento faz corresponder o seu pseudoinverso. Na tese us´amos,com frequ^encia,as seguintes igualdades, que iii s~aov´alidasem todos os epigrupos (Shevrin, 2005): x0xx0 = x0; (1) xx0 = x0x; (2) x000 = x0; (3) xx0x = x00; (4) (xy)0x = x(yx)0; (5) (xp)0 = (x0)p ; (6) para qualquer p primo. Para cada n 2 N, seja En a variedade (classe equacional) de todos os 0 n+1 0 n semigrupos un´arios(S; ·; ) que satisfazem (2.2), (2.3) e x x = x . Cada En ´euma variedade de epigrupos, e as inclus~oes En ⊂ En+1 s~aov´alidas para todos os n. Todo o semigrupo finito est´acontido em algum En e E1 ´ea variedade de semigrupos completamente regulares. Demonstr´amosum Lema, para usar mais tarde, que nos diz que para cada n 2 N, a variedade En ´eprecisamente a variedade de semigrupos un´ariosque satisfazem as identidades referidas. Variantes de semigrupos completamente regulares n~ao s~ao,em geral, com- pletamente regulares; por exemplo, se um semigrupo completamente regular tiver um zero 0, o variante em 0 ser´aum semigrupo nulo, que nem ´eregu- lar. Esta dificuldade foi contornada, em Ara´ujoet al. (2017), introduzindo a seguinte classe W de semigrupos: S 2 W () xy ´ecompletamente regular, para todo x; y 2 S: Equivalentemente, W consiste em todos os semigrupos S, de modo que o sub- semigrupo S2 = fab j a; b 2 Sg ´ecompletamente regular. A classe W inclui todos os semigrupos completamente regulares e todos os semigrupos nulos (semigrupos satisfazendo xy = uv para todos os x; y; u; v). Na Proposi¸c~ao2 resumimos os resultados relevantes, para esta tese, de Ara´ujoet al. (2017): (Proposi¸c~ao4.14) W ´ea variedade de epigrupos em E2 satisfazendo a igual- dade adicional (xy)00 = xy; (Teorema 4.15) Se S ´eum epigrupo em W, ent~ao ∼p ´etransitiva em S; (Teorema 4.17) Todo o variante de um semigrupo completamente regular est´aem W e (Corol´ario4.18) Se S ´eum variante de um semigrupo completamente regular, ent~ao ∼p ´etransitiva em S.O Teorema 4.15 tamb´emcomparava ∼p com outras no¸c~oesde conjuga¸c~ao.Na forma simplificada aqui escrita, o resultado segue facilmente do teorema de Kudryavtseva (2006): se a ∼p b, b ∼p c, e a 6= b 6= c 6= a, ent~aoexiste x; y; u; v 2 S tal que a = xy, b = yx = uv e c = vu. Assim a; b; c 2 W s~ao completamente regulares, logo a ∼p c. iv Melhor´amosum pouco a Proposi¸c~ao4.14 dizendo que a variedade W ´e precisamente a variedade de semigrupos un´ariosque satisfazem as identidades (2.2), (2.3), (2.5), (2.7) (para p = 2) e (xy)00 = xy. A variedade W tem outra caracteriza¸c~aoque n~aofoi mencionada em Ara´ujoet al. (2017). Sendo S um semigrupo, as seguintes afirma¸c~oess~aoequivalentes: S ´eum epigrupo em W, para cada c 2 S, o ideal principal esquerdo Sc ´eum subsemigrupo completamente regular e para cada c 2 S, o ideal principal direito cS ´eum subsemigrupo completamente regular. Em vista do Lema 4, tamb´emdevemos mencionar o estudo semelhante, em Liu, Chen & Han (2016), dos epigrupos S, no qual todo o submon´oide local eSe ´ecompletamente regular. A ferramenta chave na demonstra¸c~aoda Proposi¸c~ao2 (3) foi a seguinte opera¸c~aoun´aria: x∗ = (xc)0x(cx)0 : (*) 0 ∗ De facto, se (S; ·; ) ´ecompletamente regular, ent~ao(S; ·c; ) ´eum epigrupo na variedade W. No entanto, (*) foi introduzida em Ara´ujoet al. (2017) de forma \ad hoc". Para mostrar que ´ebastante natural, observamos que o ideal de um epigrupo ´eum subepigrupo (Shevrin1994). Em particular, para cada c num epigrupo S, Sc ´eum subepigrupo. Assim, para qualquer x 2 S, o pseudoinverso (xc)0 deve ter a forma yc para algum y 2 S. E´ exactamente isso que (*) faz. Seja S um epigrupo e fixemos c 2 S. Para todo x 2 S, (xc)0 = x∗c : (7) 0 ∗ Se (S; ·; ) ´eum epigrupo vamos referirmo-nos a (S; ·c; ) como o variante 0 0 un´ariode (S; ·; ) em c. A Proposi¸c~ao2 (3) afirma que, se (S; ·; ) 2 E1 ´e ∗ completamente regular, ent~ao(S; ·c; ) 2 W. O nosso primeiro resultado principal ir´amelhorar e estender isso. Primeiro temos que introduzir uma fam´ıliade variedades de semigrupos un´arios.Para cada n 2 N, a variedade Vn ´edefinida pela associatividade e pelas seguintes identidades: (2.2), (2.3), xy ··· yy00 = xy ··· y (8) | {z } | {z } n−1 n x00x ··· xy = x ··· xy (9) | {z } | {z } n−1 n Definindo y = x em, digamos, (2.9), vemos pelo Lema 1 que Vn ´euma variedade de epigrupos e, em particular, En ⊆ Vn ⊆ En+1 : (10) E´ f´acilverificar que todo o variante de um epigrupo ´eum epigrupo, mas o que n~ao´et~ao´obvio ´eo que acontece com a opera¸c~aopseudoinversa. O nosso v primeiro resultado principal esclarece isso e tamb´emo papel das variedades 0 ∗ Vn. Seja (S; ·; ) um epigrupo. Para cada c 2 S, o variante un´ario(S; ·c; ) 0 ´eum epigrupo. Se (S; ·; ) 2 Vn para algum n > 0, ent~ao(S; ·c; ∗) 2 Vn. Portanto para n 2 N, a variedade Vn ´efechada para variantes. Assim, seja (S; ·; 0) um semigrupo completamente regular. Para cada c 2 S, o variante ∗ un´ario(S; ·c; ) est´aem V1. Observ´amosque para um elemento c de um semigrupo S, a aplica¸c~ao ρc : S ! Sc; x 7! xc ´eum homomorfismo do variante (S; ·c) em (Sc; ·) pois (x·c y)c = xc·yc. Se S tamb´em´eum epigrupo, como j´aobserv´amos,o mesmo acontece com Sc. Todo o homomorfismo de semigrupos entre epigrupos ´eum homomorfismo de epigrupos, mas (2.8) mostra mais explicitamente como ρc preserva pseudoinversos. Comparar a Proposi¸c~ao2(3) com o Corol´ario7 levanta a quest~aode como as variedades V1 e W est~aorelacionadas, al´emdo facto de que ambas cont^em E1. O nosso segundo resultado principal aborda isso e o seu corol´arioliga esta 0 discuss~ao`atransitividade de ∼p. Assim, V1 ⊂ W e se (S; ·; ) ´eum epigrupo ∗ em W, ent~ao,para cada c 2 S, o variante un´ario(S; ·c; ) est´aem V1. Em particular, todo o variante de um epigrupo em W na verdade est´a numa subvariedade adequada de W, uma declara¸c~aomais forte do que a afirma¸c~aode que W ´efechado para variantes. Obtiv´emosassim, uma resposta definitiva ao Problema 6.18 em Ara´ujo et al. (2017). Como ∼p ´etransitiva em W, o teorema implica imediatamente que a conjuga¸c~aoprim´aria ∼p ´etransitiva em todos os variantes de qualquer epigrupo em W. O objectivo do terceiro cap´ıtulodesta tese foi provar o Teorema 11. Seja n > 1 um n´umerointeiro e S um semigrupo satisfazendo o seguinte: para todo x; y 2 S; xy 2 fyx; (xy)ng : Ent~aoa conjuga¸c~aoprim´aria ∼p ´etransitiva em S: Existem v´ariasmotiva¸c~oespara estudar esta classe espec´ıficade semigru- pos. Primeiro, ela naturalmente generaliza duas classes de semigrupos em que ∼p ´etransitiva: seja S um semigrupo. Se S ´ecomutativo, ent~ao ∼p ´e 2 transitiva e se S satisfaz xy = (xy) para todo x; y 2 S, ent~ao ∼p ´etransitiva. A outra motiva¸c~aopara estudar esta classe de semigrupos ´eque ela tem sido de interesse recente em outros contextos. Em particular, J. P.
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