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