On lexicographically shellable posets

Abstract

Lexicographically shellable partially ordered sets are studied. A new recursive formulation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="CL"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>CL</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {CL}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-shellability is introduced and exploited. It is shown that face lattices of convex polytopes, totally semimodular posets, posets of injective and normal words and lattices of bilinear forms are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="CL"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>CL</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {CL}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-shellable. Finally, it is shown that several common operations on graded posets preserve shellability and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="CL"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>CL</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {CL}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-shellability.