On subquasivariety lattices of semi-primal varieties. (English) Zbl 0561.08004
It is well known that the subvariety lattice of a semi-primal variety is distributive. In the present note the author constructs a sequence \(\{A_ n\}\) of semi-primal algebras with the following property: Let \(Q(A_ n)\) be the subquasivariety lattice of the variety generated by \(A_ n\). It is shown there is no non-trivial lattice identity which holds in all of the \(Q(A_ n)\), and hence there is no non-trivial lattice identity which holds in all subquasivariety lattices of semi- primal algebras.
Reviewer: A.F.Pixley
MSC:
08C15 | Quasivarieties |
08B15 | Lattices of varieties |
08A40 | Operations and polynomials in algebraic structures, primal algebras |
References:
[1] | W.Dziobiak,On distributivity of the lattice of subquasivarieties of a locally finite semi-simple arithmetical variety, to appear in Algebra Universalis. · Zbl 0548.08005 |
[2] | R. Freese andJ. B. Nation,Congruence lattices of semilattices, Pacific Journal of Mathematics49 (1973), 51-58. · Zbl 0287.06002 |
[3] | R. McNaughton,A theorem about infinite-valued sentential logic, Journal of Symbolic Logic16 (1951), 1-13. · Zbl 0043.00901 · doi:10.2307/2268660 |
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