A visual approach to test lattices

Gábor Czédli

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2009)

  • Volume: 48, Issue: 1, page 33-52
  • ISSN: 0231-9721

Abstract

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Let p be a k -ary lattice term. A k -pointed lattice L = ( L ; ∨ , ∧ , d 1 , ... , d k ) will be called a p -lattice (or a test lattice if p is not specified), if ( L ; ∨ , ∧ ) is generated by { d 1 , ... , d k } and, in addition, for any k -ary lattice term q satisfying p ( d 1 , ... , d k ) ≤ q ( d 1 , ... , d k ) in L , the lattice identity p ≤ q holds in all lattices. In an elementary visual way, we construct a finite p -lattice L ( p ) for each p . If p is a canonical lattice term, then L ( p ) coincides with the optimal p -lattice of Freese, Ježek and Nation [Freese, R., Ježek, J., Nation, J. B.: Free lattices. American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs 42, 1995, viii+293 pp.]. Some results on test lattices and short proofs for known facts on free lattices indicate that our approach is useful.

How to cite

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Czédli, Gábor. "A visual approach to test lattices." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 48.1 (2009): 33-52. <http://eudml.org/doc/35185>.

@article{Czédli2009,
abstract = {Let $p$ be a $k$-ary lattice term. A $k$-pointed lattice $L=(L;\vee ,\wedge $, $d_1,\ldots ,d_k)$ will be called a $p$-lattice (or a test lattice if $p$ is not specified), if $(L;\vee ,\wedge )$ is generated by $\lbrace d_1,\ldots ,d_k\rbrace $ and, in addition, for any $k$-ary lattice term $q$ satisfying $p(d_1,\ldots ,d_k)$$\le $$q(d_1$, $\ldots , d_k)$ in $L$, the lattice identity $p\le q$ holds in all lattices. In an elementary visual way, we construct a finite $p$-lattice $L(p)$ for each $p$. If $p$ is a canonical lattice term, then $L(p)$ coincides with the optimal $p$-lattice of Freese, Ježek and Nation [Freese, R., Ježek, J., Nation, J. B.: Free lattices. American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs 42, 1995, viii+293 pp.]. Some results on test lattices and short proofs for known facts on free lattices indicate that our approach is useful.},
author = {Czédli, Gábor},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Free lattice; test lattice; lattice identity; Whitman’s condition; free lattice; test lattice; lattice identity; Whitman's condition},
language = {eng},
number = {1},
pages = {33-52},
publisher = {Palacký University Olomouc},
title = {A visual approach to test lattices},
url = {http://eudml.org/doc/35185},
volume = {48},
year = {2009},
}

TY - JOUR
AU - Czédli, Gábor
TI - A visual approach to test lattices
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2009
PB - Palacký University Olomouc
VL - 48
IS - 1
SP - 33
EP - 52
AB - Let $p$ be a $k$-ary lattice term. A $k$-pointed lattice $L=(L;\vee ,\wedge $, $d_1,\ldots ,d_k)$ will be called a $p$-lattice (or a test lattice if $p$ is not specified), if $(L;\vee ,\wedge )$ is generated by $\lbrace d_1,\ldots ,d_k\rbrace $ and, in addition, for any $k$-ary lattice term $q$ satisfying $p(d_1,\ldots ,d_k)$$\le $$q(d_1$, $\ldots , d_k)$ in $L$, the lattice identity $p\le q$ holds in all lattices. In an elementary visual way, we construct a finite $p$-lattice $L(p)$ for each $p$. If $p$ is a canonical lattice term, then $L(p)$ coincides with the optimal $p$-lattice of Freese, Ježek and Nation [Freese, R., Ježek, J., Nation, J. B.: Free lattices. American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs 42, 1995, viii+293 pp.]. Some results on test lattices and short proofs for known facts on free lattices indicate that our approach is useful.
LA - eng
KW - Free lattice; test lattice; lattice identity; Whitman’s condition; free lattice; test lattice; lattice identity; Whitman's condition
UR - http://eudml.org/doc/35185
ER -

References

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  6. Freese, R., Ježek, J., Nation, J. B., Free lattices, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs 42, 1995, viii+293 pp. (1995) MR1319815
  7. Freese, R., Nation, J. B., Congruence lattices of semilattices, Pacific J. Math. 49 (1973), 51–58. (1973) Zbl0287.06002MR0332590
  8. Grätzer, G., General Lattice Theory, Birkhäuser Verlag, 1998 sec. ed. (1998) MR1670580
  9. Haiman, M., Proof theory for linear lattices, Adv. in Math. 58 (1985), 209–242. (1985) Zbl0584.06003MR0815357
  10. Jónsson, B., On the representation of lattices, Math. Scandinavica 1 (1953), 193–206. (1953) MR0058567
  11. Jónsson, B., Algebras whose congruence lattices are distributive, Math. Scandinavica 21 (1967), 110–121. (1967) MR0237402
  12. Lipparini, P., From congruence identities to tolerance identities, Acta Sci. Math. (Szeged) 73 (2007), 31–51. (2007) Zbl1136.08001MR2339851
  13. Whitman, Ph. M., Free lattices, Ann. of Math. 42 (1941), 325–330. (1941) Zbl0024.24501MR0003614

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