# Article

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Keywords:
Free lattice; test lattice; lattice identity; Whitman’s condition
Summary:
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.
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