Finite basis theorems for relatively congruence-distributive quasivarieties

Abstract

<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is any quasivariety. A congruence relation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Theta"> <mml:semantics> <mml:mi mathvariant="normal"> Θ </mml:mi> <mml:annotation encoding="application/x-tex">\Theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a member <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf {A}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> - <italic>congruence</italic> if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper A slash normal upper Theta element-of script upper Q"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi mathvariant="normal"> Θ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf {A}}/\Theta \in \mathcal {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C o n Subscript script upper Q Baseline bold upper A"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>o</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Q</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">Co{n_\mathcal {Q}}{\mathbf {A}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -congruences is closed under arbitrary intersection and hence forms a complete lattice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper C bold o bold n Subscript script upper Q Baseline bold upper A"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">C</mml:mi> <mml:mi mathvariant="bold">o</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">n</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Q</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf {Co}}{{\mathbf {n}}_\mathcal {Q}}{\mathbf {A}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>relatively congruence-distributive</italic> if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper C bold o bold n Subscript script upper Q Baseline bold upper A"> <mml:semantics>