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Quasivarieties of Sugihara semilattices with involution. (English. Russian original) Zbl 0954.06003
Algebra Logika 39, No. 1, 47-65 (2000); translation in Algebra Logic 39, No. 1, 26-36 (2000).
It is shown that the lattice of quasivarieties contained in the quasivariety generated by an $$n$$-element Sugihara semilattice with involution (i.e. $$\{\wedge, ^{\overline{\;}}\}$$-algebras that are term equivalent to subalgebras of $$\{\rightarrow, ^{\overline{\;}}\}$$-reducts of Sugihara algebras) includes a sublattice isomorphic to the ideal lattice of a free lattice with $$\omega$$ free generators if and only if $$n \geqslant 3$$. In particular, the above-mentioned lattice of quasivarieties is of cardinality $$2^{\aleph_{0}}$$ if and only if $$n \geqslant 3$$.

##### MSC:
 06A12 Semilattices 08C15 Quasivarieties 06B05 Structure theory of lattices 06B25 Free lattices, projective lattices, word problems
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