## 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.

### MSC:

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

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