Discussiones Mathematicae General Algebra and Applications 24(2) (2004) 267-275
doi: 10.7151/dmgaa.1089

[BIBTex] [PDF] [PS]

ON VARIETIES OF LEFT DISTRIBUTIVE LEFT IDEMPOTENT GROUPOIDS

David Stanovský

Charles University in Prague,
KA MFF UK, Sokolovská 83,
18675 Praha, Czech Republic

e-mail: stanovsk@karlin.mff.cuni.cz

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.

Keywords: left distributivity, left idempotence, right zero band, LDLI groupoids, subdirectly irreducible, free groupoid, lattice of subvarieties.

2000 Mathematics Subject Classification: 08B15, 08B20, 08B26, 20N02.

References

[1] G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc. 31 (1935), 433-454.
[2]S. Burris and H.P. Sankappanavar, A course in universal algebra, Springer, New York 1981 (and also the (electronic) Millennium Edition 1999).
[3] R. Fenn and C. Rourke, Racks and links in codimension two, J. Knot Theory Ramifications 1 (1992), 343-406.
[4] P. Jedlicka, On left distributive left idempotent groupoids, Comment. Math. Univ. Carolinae, to appear.
[5] T. Kepka, Non-idempotent left symmetric left distributive groupoids, Comment. Math. Univ. Carolinae 35 (1994), 181-186.
[6] J. Płonka, On k-cyclic groupoids, Math. Japon. 30 (1985), 371-382.
[7] B. Roszkowska, The lattice of varieties of symmetric idempotent entropic groupoids, Demonstratio Math. 20 (1987), 259-275.
[8] H. Ryder, The congruence structure of racks, Comm. Algebra 23 (1995), 4971-4989.
[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

Received 13 August 2004
Revised 30 December 2004