Geodesic
Relations on a lattice of varieties of completely regular semigroups
Mathematica Bohemica, Tome 145 (2020) no. 3, pp. 225-240
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Completely regular semigroups $\mathcal {CR}$ are considered here with the unary operation of inversion within the maximal subgroups of the semigroup. This makes $\mathcal {CR}$ a variety; its lattice of subvarieties is denoted by $\mathcal {L(CR)}$. We study here the relations ${\mathbf K,T,L}$ and ${\mathbf C}$ relative to a sublattice $\Psi $ of $\mathcal {L(CR)}$ constructed in a previous publication. \endgraf For ${\mathbf R}$ being any of these relations, we determine the ${\mathbf R}$-classes of all varieties in the lattice $\Psi $ as well as the restrictions of ${\mathbf R}$ to $\Psi $.
Completely regular semigroups $\mathcal {CR}$ are considered here with the unary operation of inversion within the maximal subgroups of the semigroup. This makes $\mathcal {CR}$ a variety; its lattice of subvarieties is denoted by $\mathcal {L(CR)}$. We study here the relations ${\mathbf K,T,L}$ and ${\mathbf C}$ relative to a sublattice $\Psi $ of $\mathcal {L(CR)}$ constructed in a previous publication. \endgraf For ${\mathbf R}$ being any of these relations, we determine the ${\mathbf R}$-classes of all varieties in the lattice $\Psi $ as well as the restrictions of ${\mathbf R}$ to $\Psi $.
DOI : 10.21136/MB.2019.0050-18
Classification : 20M07
Keywords: semigroup; completely regular; variety; lattice; relation; kernel; trace; local relation; core 
@article{10_21136_MB_2019_0050_18,
     author = {Petrich, Mario},
     title = {Relations on a lattice of varieties of completely regular semigroups},
     journal = {Mathematica Bohemica},
     pages = {225--240},
     year = {2020},
     volume = {145},
     number = {3},
     doi = {10.21136/MB.2019.0050-18},
     mrnumber = {4221831},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2019.0050-18/}
}
TY  - JOUR
AU  - Petrich, Mario
TI  - Relations on a lattice of varieties of completely regular semigroups
JO  - Mathematica Bohemica
PY  - 2020
SP  - 225
EP  - 240
VL  - 145
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2019.0050-18/
DO  - 10.21136/MB.2019.0050-18
LA  - en
ID  - 10_21136_MB_2019_0050_18
ER  - 
%0 Journal Article
%A Petrich, Mario
%T Relations on a lattice of varieties of completely regular semigroups
%J Mathematica Bohemica
%D 2020
%P 225-240
%V 145
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2019.0050-18/
%R 10.21136/MB.2019.0050-18
%G en
%F 10_21136_MB_2019_0050_18
Petrich, Mario. Relations on a lattice of varieties of completely regular semigroups. Mathematica Bohemica, Tome 145 (2020) no. 3, pp. 225-240. doi: 10.21136/MB.2019.0050-18

[1] Kad'ourek, J.: On the word problem for free bands of groups and for free objects in some other varieties of completely regular semigroups. Semigroup Forum 38 (1989), 1-55. | DOI | MR | JFM

[2] Pastijn, F.: The lattice of completely regular semigroup varieties. J. Aust. Math. Soc., Ser. A 49 (1990), 24-42. | DOI | MR | JFM

[3] Petrich, M.: Some relations on the lattice of varieties of completely regular semigroups. Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat., VIII. Ser. 5 (2002), 265-278. | MR | JFM

[4] Petrich, M.: Canonical varieties of completely regular semigroups. J. Aust. Math. Soc. 83 (2007), 87-104. | DOI | MR | JFM

[5] Petrich, M.: A lattice of varieties of completely regular semigroups. Commun. Algebra 42 (2014), 1397-1413. | DOI | MR | JFM

[6] Petrich, M.: Certain relations on a lattice of varieties of completely regular semigroups. Commun. Algebra 43 (2015), 4080-4096. | DOI | MR | JFM

[7] Petrich, M.: Varieties of completely regular semigroups related to canonical varieties. Semigroup Forum 90 (2015), 53-99. | DOI | MR | JFM

[8] Petrich, M.: Some relations on a semilattice of varieties of completely regular semigroups. Semigroup Forum 93 (2016), 607-628. | DOI | MR | JFM

[9] Petrich, M.: Another lattice of varieties of completely regular semigroups. Commun. Algebra 45 (2017), 2783-2794. | DOI | MR | JFM

[10] Petrich, M., Reilly, N. R.: Semigroups generated by certain operators on varieties of completely regular semigroups. Pac. J. Math. 132 (1988), 151-175. | DOI | MR | JFM

[11] Petrich, M., Reilly, N. R.: Operators related to $E$-disjunctive and fundamental completely regular semigroups. J. Algebra 134 (1990), 1-27. | DOI | MR | JFM

[12] Petrich, M., Reilly, N. R.: Operators related to idempotent generated and monoid completely regular semigroups. J. Aust. Math. Soc., Ser. A 49 (1990), 1-23. | DOI | MR | JFM

[13] Petrich, M., Reilly, N. R.: Completely Regular Semigroups. Canadian Mathematical Society Series of Monographs and Advanced Texts 23. A Wiley-Interscience Publication. John Wiley & Sons, Chichester (1999). | MR | JFM

[14] Polák, L.: On varieties of completely regular semigroups. I. Semigroup Forum 32 (1985), 97-123. | DOI | MR | JFM

[15] Polák, L.: On varieties of completely regular semigroups. II. Semigroup Forum 36 (1988), 253-284. | DOI | MR | JFM

[16] Reilly, N. R., Zhang, S.: Decomposition of the lattice of pseudovarieties of finite semigroups induced by bands. Algebra Univers. 44 (2000), 217-239. | DOI | MR | JFM

Cité par Sources :