Geodesic
A visual approach to test lattices
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 48 (2009) no. 1, pp. 33-52 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 06B25
Keywords: Free lattice; test lattice; lattice identity; Whitman’s condition 
@article{AUPO_2009_48_1_a3,
     author = {Cz\'edli, G\'abor},
     title = {A visual approach to test lattices},
     journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
     pages = {33--52},
     year = {2009},
     volume = {48},
     number = {1},
     mrnumber = {2641946},
     zbl = {1203.06011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUPO_2009_48_1_a3/}
}
TY  - JOUR
AU  - Czédli, Gábor
TI  - A visual approach to test lattices
JO  - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY  - 2009
SP  - 33
EP  - 52
VL  - 48
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/AUPO_2009_48_1_a3/
LA  - en
ID  - AUPO_2009_48_1_a3
ER  - 
%0 Journal Article
%A Czédli, Gábor
%T A visual approach to test lattices
%J Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
%D 2009
%P 33-52
%V 48
%N 1
%U http://geodesic.mathdoc.fr/item/AUPO_2009_48_1_a3/
%G en
%F AUPO_2009_48_1_a3
Czédli, Gábor. A visual approach to test lattices. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 48 (2009) no. 1, pp. 33-52. http://geodesic.mathdoc.fr/item/AUPO_2009_48_1_a3/