Finite sublattices in the lattice of clones. (English. Russian original) Zbl 0838.06006

Algebra Logic 33, No. 5, 287-306 (1994); translation from Algebra Logika 33, No. 5, 514-549 (1994).
Summary: The lattice \({\mathcal L}_k\) of clones of functions over a \(k\)-element set is studied. It is shown that every lattice which is a countable direct product of finite lattices is embedded (up to isomorphism) in \({\mathcal L}_4\) and, hence, in \({\mathcal L}_k\) for \(k\geq 4\). This directly implies that every finite and any countable residually finite lattice is embedded in \({\mathcal L}_k\), \(k\geq 4\), and that no nontrivial quasi-identity holds in \({\mathcal L}_k\), \(k\geq 4\). A number of particular lattices (which are free in some lattice varieties) embeddable in \({\mathcal L}_k\), \(k\geq 4\), are presented.


06B15 Representation theory of lattices
06B25 Free lattices, projective lattices, word problems
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