  In this note, we investigate some of the fundamental algebraic and geometric
properties of $s$-lecture hall simplices and their generalizations. We show
that all $s$-lecture hall order polytopes, which simultaneously generalize
$s$-lecture hall simplices and order polytopes, satisfy a property which
implies the integer decomposition property. This answers one conjecture of
Hibi, Olsen and Tsuchiya. By relating $s$-lecture hall polytopes to alcoved
polytopes, we then use this property to show that families of $s$-lecture hall
simplices admit a quadratic Gr\"obner basis with a square-free initial ideal.
Consequently, we find that all $s$-lecture hall simplices for which the first
order difference sequence of $s$ is a $0,1$-sequence have a regular and
unimodular triangulation. This answers a second conjecture of Hibi, Olsen and
Tsuchiya, and it gives a partial answer to a conjecture of Beck, Braun,
K\"oppe, Savage and Zafeirakopoulos.
