Independent axiom systems for nearlattices

João Araújo; Michael Kinyon

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 4, page 975-992
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Araújo, João, and Kinyon, Michael. "Independent axiom systems for nearlattices." Czechoslovak Mathematical Journal 61.4 (2011): 975-992. <http://eudml.org/doc/196329>.

@article{Araújo2011,
abstract = {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.},
author = {Araújo, João, Kinyon, Michael},
journal = {Czechoslovak Mathematical Journal},
keywords = {nearlattice; equational base; nearlattice; join semilattice; axiom system; Prover9},
language = {eng},
number = {4},
pages = {975-992},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Independent axiom systems for nearlattices},
url = {http://eudml.org/doc/196329},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Araújo, João
AU - Kinyon, Michael
TI - Independent axiom systems for nearlattices
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 4
SP - 975
EP - 992
AB - 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.
LA - eng
KW - nearlattice; equational base; nearlattice; join semilattice; axiom system; Prover9
UR - http://eudml.org/doc/196329
ER -

References

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  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. (2006) MR2353310
  2. Chajda, I., Halaš, R., Kühr, J., Semilattice structures, Research and Exposition in Mathematics 30, Heldermann Verlag, Lemgo (2007). (2007) MR2326262
  3. Chajda, I., Kolařík, M., 10.1016/j.disc.2007.09.009, Discrete Math. 308 (2008), 4906-4913. (2008) MR2446101DOI10.1016/j.disc.2007.09.009
  4. Hickman, R., 10.1080/00927878008822537, Commun. Algebra 8 (1980), 1653-1685. (1980) Zbl0436.06003MR0585925DOI10.1080/00927878008822537
  5. McCune, W., Prover9 and Mace4, version 2009-11A, (http://www.cs.unm.edu/mccune/prover9/). 

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