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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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