On varieties of left distributive left idempotent groupoids

David Stanovský

Discussiones Mathematicae - General Algebra and Applications (2004)

  • Volume: 24, Issue: 2, page 267-275
  • ISSN: 1509-9415

Abstract

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We describe a part of the lattice of subvarieties of left distributive left idempotent groupoids (i.e. those satisfying the identities x(yz) ≈ (xy)(xz) and (xx)y ≈ xy) modulo the lattice of subvarieties of left distributive idempotent groupoids. A free groupoid in a subvariety of LDLI groupoids satisfying an identity xⁿ ≈ x decomposes as the direct product of its largest idempotent factor and a cycle. Some properties of subdirectly ireducible LDLI groupoids are found.

How to cite

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David Stanovský. "On varieties of left distributive left idempotent groupoids." Discussiones Mathematicae - General Algebra and Applications 24.2 (2004): 267-275. <http://eudml.org/doc/287714>.

@article{DavidStanovský2004,
abstract = {We describe a part of the lattice of subvarieties of left distributive left idempotent groupoids (i.e. those satisfying the identities x(yz) ≈ (xy)(xz) and (xx)y ≈ xy) modulo the lattice of subvarieties of left distributive idempotent groupoids. A free groupoid in a subvariety of LDLI groupoids satisfying an identity xⁿ ≈ x decomposes as the direct product of its largest idempotent factor and a cycle. Some properties of subdirectly ireducible LDLI groupoids are found.},
author = {David Stanovský},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {left distributivity; left idempotence; right zero band; LDLI groupoids; subdirectly irreducible; free groupoid; lattice of subvarieties},
language = {eng},
number = {2},
pages = {267-275},
title = {On varieties of left distributive left idempotent groupoids},
url = {http://eudml.org/doc/287714},
volume = {24},
year = {2004},
}

TY - JOUR
AU - David Stanovský
TI - On varieties of left distributive left idempotent groupoids
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2004
VL - 24
IS - 2
SP - 267
EP - 275
AB - We describe a part of the lattice of subvarieties of left distributive left idempotent groupoids (i.e. those satisfying the identities x(yz) ≈ (xy)(xz) and (xx)y ≈ xy) modulo the lattice of subvarieties of left distributive idempotent groupoids. A free groupoid in a subvariety of LDLI groupoids satisfying an identity xⁿ ≈ x decomposes as the direct product of its largest idempotent factor and a cycle. Some properties of subdirectly ireducible LDLI groupoids are found.
LA - eng
KW - left distributivity; left idempotence; right zero band; LDLI groupoids; subdirectly irreducible; free groupoid; lattice of subvarieties
UR - http://eudml.org/doc/287714
ER -

References

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  2. [2] S. Burris and H.P. Sankappanavar, A course in universal algebra, Springer, New York 1981 (and also the (electronic) Millennium Edition 1999). Zbl0478.08001
  3. [3] R. Fenn and C. Rourke, Racks and links in codimension two, J. Knot Theory Ramifications 1 (1992), 343-406. Zbl0787.57003
  4. [4] P. Jedlicka, On left distributive left idempotent groupoids, Comment. Math. Univ. Carolinae, to appear. Zbl1106.20049
  5. [5] T. Kepka, Non-idempotent left symmetric left distributive groupoids, Comment. Math. Univ. Carolinae 35 (1994), 181-186. Zbl0807.20057
  6. [6] J. Płonka, On k-cyclic groupoids, Math. Japon. 30 (1985), 371-382. Zbl0572.08004
  7. [7] B. Roszkowska, The lattice of varieties of symmetric idempotent entropic groupoids, Demonstratio Math. 20 (1987), 259-275. Zbl0663.20065
  8. [8] H. Ryder, The congruence structure of racks, Comm. Algebra 23 (1995), 4971-4989. Zbl0842.57011
  9. [9] D. Stanovský, Left distributive left quasigroups, PhD Thesis, Charles University in Prague, 2004, Available at http://www.karlin.mff.cuni.cz/~stanovsk/math/disert.pdf. Zbl1076.08004

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