Bases for certain varieties of completely regular semigroups

Mario Petrich

Commentationes Mathematicae Universitatis Carolinae (2021)

  • Issue: 1, page 41-65
  • ISSN: 0010-2628

Abstract

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Completely regular semigroups equipped with the unary operation of inversion within their maximal subgroups form a variety, denoted by 𝒞ℛ . The lattice of subvarieties of 𝒞ℛ is denoted by ℒ ( 𝒞ℛ ) . For each variety in an ⋂ -subsemilattice Γ of ℒ ( 𝒞ℛ ) , we construct at least one basis of identities, and for some important varieties, several. We single out certain remarkable types of bases of general interest. As an application for the local relation L , we construct 𝐋 -classes of all varieties in Γ . Two figures illustrate the theory.

How to cite

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Petrich, Mario. "Bases for certain varieties of completely regular semigroups." Commentationes Mathematicae Universitatis Carolinae (2021): 41-65. <http://eudml.org/doc/297849>.

@article{Petrich2021,
abstract = {Completely regular semigroups equipped with the unary operation of inversion within their maximal subgroups form a variety, denoted by $\mathcal \{CR\}$. The lattice of subvarieties of $\,\mathcal \{CR\}$ is denoted by $\mathcal \{L\}(\mathcal \{CR\})$. For each variety in an $\bigcap $-subsemilattice $\Gamma $ of $\mathcal \{L\}(\mathcal \{CR\})$, we construct at least one basis of identities, and for some important varieties, several. We single out certain remarkable types of bases of general interest. As an application for the local relation $L$, we construct $\mathbf \{L\}$-classes of all varieties in $\Gamma $. Two figures illustrate the theory.},
author = {Petrich, Mario},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {semigroup; completely regular; variety; basis; local relation},
language = {eng},
number = {1},
pages = {41-65},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Bases for certain varieties of completely regular semigroups},
url = {http://eudml.org/doc/297849},
year = {2021},
}

TY - JOUR
AU - Petrich, Mario
TI - Bases for certain varieties of completely regular semigroups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2021
PB - Charles University in Prague, Faculty of Mathematics and Physics
IS - 1
SP - 41
EP - 65
AB - Completely regular semigroups equipped with the unary operation of inversion within their maximal subgroups form a variety, denoted by $\mathcal {CR}$. The lattice of subvarieties of $\,\mathcal {CR}$ is denoted by $\mathcal {L}(\mathcal {CR})$. For each variety in an $\bigcap $-subsemilattice $\Gamma $ of $\mathcal {L}(\mathcal {CR})$, we construct at least one basis of identities, and for some important varieties, several. We single out certain remarkable types of bases of general interest. As an application for the local relation $L$, we construct $\mathbf {L}$-classes of all varieties in $\Gamma $. Two figures illustrate the theory.
LA - eng
KW - semigroup; completely regular; variety; basis; local relation
UR - http://eudml.org/doc/297849
ER -

References

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  5. Petrich M., 10.1007/s00233-014-9591-2, Semigroup Forum 90 (2015), no. 1, 53–99. MR3297810DOI10.1007/s00233-014-9591-2
  6. Petrich M., 10.21136/MB.2018.0112-17, Math. Bohem. 145 (2020), no. 1, 1–14. MR4088688DOI10.21136/MB.2018.0112-17
  7. Petrich M., Relations on some varieties of completely regular semigroups, manuscript. 
  8. Petrich M., Reilly N. R., 10.1016/0021-8693(90)90207-5, J. Algebra 134 (1990), no. 1, 1–27. MR1068411DOI10.1016/0021-8693(90)90207-5
  9. Petrich M., Reilly N. R., 10.1017/S1446788700030202, J. Austral. Math. Soc. Ser. A 49 (1990), no. 1, 1–23. MR1054079DOI10.1017/S1446788700030202
  10. 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, New York, 1999. MR1684919
  11. Reilly N. R., 10.1017/S144678870002365X, J. Austral. Math. Soc. Ser. A 38 (1985), no. 3, 372–393. MR0779201DOI10.1017/S144678870002365X

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