Definability for equational theories of commutative groupoids

Jaroslav Ježek

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 2, page 305-333
  • ISSN: 0011-4642

Abstract

top
We find several large classes of equations with the property that every automorphism of the lattice of equational theories of commutative groupoids fixes any equational theory generated by such equations, and every equational theory generated by finitely many such equations is a definable element of the lattice. We conjecture that the lattice has no non-identical automorphisms.

How to cite

top

Ježek, Jaroslav. "Definability for equational theories of commutative groupoids." Czechoslovak Mathematical Journal 62.2 (2012): 305-333. <http://eudml.org/doc/247133>.

@article{Ježek2012,
abstract = {We find several large classes of equations with the property that every automorphism of the lattice of equational theories of commutative groupoids fixes any equational theory generated by such equations, and every equational theory generated by finitely many such equations is a definable element of the lattice. We conjecture that the lattice has no non-identical automorphisms.},
author = {Ježek, Jaroslav},
journal = {Czechoslovak Mathematical Journal},
keywords = {simple algebra; idempotent; group; definability; equational theory; commutative groupoid; automorphisms},
language = {eng},
number = {2},
pages = {305-333},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Definability for equational theories of commutative groupoids},
url = {http://eudml.org/doc/247133},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Ježek, Jaroslav
TI - Definability for equational theories of commutative groupoids
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 2
SP - 305
EP - 333
AB - We find several large classes of equations with the property that every automorphism of the lattice of equational theories of commutative groupoids fixes any equational theory generated by such equations, and every equational theory generated by finitely many such equations is a definable element of the lattice. We conjecture that the lattice has no non-identical automorphisms.
LA - eng
KW - simple algebra; idempotent; group; definability; equational theory; commutative groupoid; automorphisms
UR - http://eudml.org/doc/247133
ER -

References

top
  1. Grech, M., 10.1016/S0021-8693(02)00674-9, J. Algebra 261 (2003), 207-228. (2003) Zbl1026.20040MR1967162DOI10.1016/S0021-8693(02)00674-9
  2. Grech, M., 10.1090/S0002-9947-09-04849-1, Trans. Am. Math. Soc. 361 (2009), 3435-3462. (2009) MR2491887DOI10.1090/S0002-9947-09-04849-1
  3. Ježek, J., The lattice of equational theories. Part I: Modular elements, Czech. Math. J. 31 (1981), 127-152. (1981) MR0604120
  4. Ježek, J., The lattice of equational theories. Part II: The lattice of full sets of terms, Czech. Math. J. 31 (1981), 573-603. (1981) MR0631604
  5. Ježek, J., The lattice of equational theories. Part III: Definability and automorphisms, Czech. Math. J. 32 (1982), 129-164. (1982) MR0646718
  6. Ježek, J., The lattice of equational theories. Part IV: Equational theories of finite algebras, Czech. Math. J. 36 (1986), 331-341. (1986) MR0831318
  7. Ježek, J., 10.1007/s10587-006-0010-z, Czech. Math. J. 56 (2006), 133-154. (2006) Zbl1164.03318MR2207011DOI10.1007/s10587-006-0010-z
  8. Ježek, J., McKenzie, R., 10.1007/BF02573566, Semigroup Forum 46 (1993), 199-245. (1993) Zbl0782.20051MR1200214DOI10.1007/BF02573566
  9. Kisielewicz, A., 10.1090/S0002-9947-03-03351-8, Trans. Am. Math. Soc. 356 (2004), 3483-3504. (2004) Zbl1050.08005MR2055743DOI10.1090/S0002-9947-03-03351-8
  10. McKenzie, R. N., McNulty, G. F., Taylor, W. F., Algebras, Lattices, Varieties. Volume I, Wadsworth & Brooks/Cole Monterey (1987). (1987) Zbl0611.08001MR0883644
  11. Tarski, A., Equational logic and equational theories of algebras. Proc. Logic Colloq., Hannover 1966, Contrib. Math. Logic (1968), 275-288. (1968) MR0237410
  12. Vernikov, B. M., Proofs of definability of some varieties and sets of varieties of semigroups, Preprint. MR2898768

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.