Subdirectly irreducible members of products of lattice varieties
Subdirectly irreducible members of products of lattice varieties
In this paper we prove: Theorem.<italic>Let</italic><inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper V"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">V</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathbf {V}</mml:annotation></mml:semantics></mml:math></inline-formula><italic>and</italic><inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper W"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">W</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathbf {W}</mml:annotation></mml:semantics></mml:math></inline-formula><italic>be nontrivial lattice varieties. If</italic><inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L element-of bold upper V ring bold upper W"><mml:semantics><mml:mrow><mml:mi>L</mml:mi><mml:mo>∈<!-- ∈ --></mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">V</mml:mi></mml:mrow><mml:mo>∘<!-- ∘ --></mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">W</mml:mi></mml:mrow></mml:mrow><mml:annotation encoding="application/x-tex">L \in …