Geodesic
Independent axiom systems for nearlattices
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 975-992

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

DOI MR   Zbl

A nearlattice is a join semilattice such that every principal filter is a lattice with respect to the induced order. Hickman and later Chajda et al independently showed that nearlattices can be treated as varieties of algebras with a ternary operation satisfying certain axioms. Our main result is that the variety of nearlattices is $2$-based, and we exhibit an explicit system of two independent identities. We also show that the original axiom systems of Hickman as well as that of Chajda et al are dependent.
A nearlattice is a join semilattice such that every principal filter is a lattice with respect to the induced order. Hickman and later Chajda et al independently showed that nearlattices can be treated as varieties of algebras with a ternary operation satisfying certain axioms. Our main result is that the variety of nearlattices is $2$-based, and we exhibit an explicit system of two independent identities. We also show that the original axiom systems of Hickman as well as that of Chajda et al are dependent.
DOI : 10.1007/s10587-011-0062-6
Classification : 06A12, 06B75, 68T15
Keywords: nearlattice; equational base 
Araújo, João; Kinyon, Michael. Independent axiom systems for nearlattices. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 975-992. doi: 10.1007/s10587-011-0062-6
@article{10_1007_s10587_011_0062_6,
     author = {Ara\'ujo, Jo\~ao and Kinyon, Michael},
     title = {Independent axiom systems for nearlattices},
     journal = {Czechoslovak Mathematical Journal},
     pages = {975--992},
     year = {2011},
     volume = {61},
     number = {4},
     doi = {10.1007/s10587-011-0062-6},
     mrnumber = {2886250},
     zbl = {1249.06003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-011-0062-6/}
}
TY  - JOUR
AU  - Araújo, João
AU  - Kinyon, Michael
TI  - Independent axiom systems for nearlattices
JO  - Czechoslovak Mathematical Journal
PY  - 2011
SP  - 975
EP  - 992
VL  - 61
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-011-0062-6/
DO  - 10.1007/s10587-011-0062-6
LA  - en
ID  - 10_1007_s10587_011_0062_6
ER  - 
%0 Journal Article
%A Araújo, João
%A Kinyon, Michael
%T Independent axiom systems for nearlattices
%J Czechoslovak Mathematical Journal
%D 2011
%P 975-992
%V 61
%N 4
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-011-0062-6/
%R 10.1007/s10587-011-0062-6
%G en
%F 10_1007_s10587_011_0062_6

[1] Chajda, I., Halaš, R.: An example of a congruence distributive variety having no near-unanimity term. Acta Univ. M. Belii Ser. Math. 13 (2006), 29-31. | MR

[2] Chajda, I., Halaš, R., Kühr, J.: Semilattice structures. Research and Exposition in Mathematics 30, Heldermann Verlag, Lemgo (2007). | MR

[3] Chajda, I., Kolařík, M.: Nearlattices. Discrete Math. 308 (2008), 4906-4913. | DOI | MR

[4] Hickman, R.: Join algebras. Commun. Algebra 8 (1980), 1653-1685. | DOI | MR | Zbl

[5] McCune, W.: Prover9 and Mace4, version 2009-11A. \hfil ( http://www.cs.unm.edu/mccune/prover9/)

Cité par Sources :