Relations on a lattice of varieties of completely regular semigroups

Mario Petrich

Mathematica Bohemica (2020)

  • Volume: 145, Issue: 3, page 225-240
  • ISSN: 0862-7959

Abstract

top
Completely regular semigroups 𝒞ℛ are considered here with the unary operation of inversion within the maximal subgroups of the semigroup. This makes 𝒞ℛ a variety; its lattice of subvarieties is denoted by ℒ ( 𝒞ℛ ) . We study here the relations 𝐊 , T , L and 𝐂 relative to a sublattice Ψ of ℒ ( 𝒞ℛ ) constructed in a previous publication. For 𝐑 being any of these relations, we determine the 𝐑 -classes of all varieties in the lattice Ψ as well as the restrictions of 𝐑 to Ψ .

How to cite

top

Petrich, Mario. "Relations on a lattice of varieties of completely regular semigroups." Mathematica Bohemica 145.3 (2020): 225-240. <http://eudml.org/doc/296970>.

@article{Petrich2020,
abstract = {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. 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 $.},
author = {Petrich, Mario},
journal = {Mathematica Bohemica},
keywords = {semigroup; completely regular; variety; lattice; relation; kernel; trace; local relation; core},
language = {eng},
number = {3},
pages = {225-240},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Relations on a lattice of varieties of completely regular semigroups},
url = {http://eudml.org/doc/296970},
volume = {145},
year = {2020},
}

TY - JOUR
AU - Petrich, Mario
TI - Relations on a lattice of varieties of completely regular semigroups
JO - Mathematica Bohemica
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 145
IS - 3
SP - 225
EP - 240
AB - 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. 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 $.
LA - eng
KW - semigroup; completely regular; variety; lattice; relation; kernel; trace; local relation; core
UR - http://eudml.org/doc/296970
ER -

References

top
  1. Kad'ourek, J., 10.1007/BF02573217, Semigroup Forum 38 (1989), 1-55. (1989) Zbl0661.20037MR0961825DOI10.1007/BF02573217
  2. Pastijn, F., 10.1017/s1446788700030214, J. Aust. Math. Soc., Ser. A 49 (1990), 24-42. (1990) Zbl0706.20042MR1054080DOI10.1017/s1446788700030214
  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. (2002) Zbl1072.20067MR1911191
  4. Petrich, M., 10.1017/S1446788700036405, J. Aust. Math. Soc. 83 (2007), 87-104. (2007) Zbl1142.20035MR2378436DOI10.1017/S1446788700036405
  5. Petrich, M., 10.1080/00927872.2012.667181, Commun. Algebra 42 (2014), 1397-1413. (2014) Zbl1302.20059MR3169638DOI10.1080/00927872.2012.667181
  6. Petrich, M., 10.1080/00927872.2014.907412, Commun. Algebra 43 (2015), 4080-4096. (2015) Zbl1339.20053MR3366561DOI10.1080/00927872.2014.907412
  7. Petrich, M., 10.1007/s00233-014-9591-2, Semigroup Forum 90 (2015), 53-99. (2015) Zbl1328.20077MR3297810DOI10.1007/s00233-014-9591-2
  8. Petrich, M., 10.1007/s00233-016-9817-6, Semigroup Forum 93 (2016), 607-628. (2016) Zbl06688592MR3572420DOI10.1007/s00233-016-9817-6
  9. Petrich, M., 10.1080/00927872.2016.1233190, Commun. Algebra 45 (2017), 2783-2794. (2017) Zbl1373.20072MR3594557DOI10.1080/00927872.2016.1233190
  10. Petrich, M., Reilly, N. R., 10.2140/pjm.1988.132.151, Pac. J. Math. 132 (1988), 151-175. (1988) Zbl0598.20061MR0929587DOI10.2140/pjm.1988.132.151
  11. Petrich, M., Reilly, N. R., 10.1016/0021-8693(90)90207-5, J. Algebra 134 (1990), 1-27. (1990) Zbl0706.20043MR1068411DOI10.1016/0021-8693(90)90207-5
  12. Petrich, M., Reilly, N. R., 10.1017/s1446788700030202, J. Aust. Math. Soc., Ser. A 49 (1990), 1-23. (1990) Zbl0708.20019MR1054079DOI10.1017/s1446788700030202
  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). (1999) Zbl0967.20034MR1684919
  14. Polák, L., 10.1007/BF02575527, Semigroup Forum 32 (1985), 97-123. (1985) Zbl0564.20034MR0803483DOI10.1007/BF02575527
  15. Polák, L., 10.1007/BF02575021, Semigroup Forum 36 (1988), 253-284. (1988) Zbl0638.20032MR0916425DOI10.1007/BF02575021
  16. Reilly, N. R., Zhang, S., 10.1007/s000120050183, Algebra Univers. 44 (2000), 217-239. (2000) Zbl1013.08010MR1816020DOI10.1007/s000120050183

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.