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This paper concerns the combinatorics of the orbit Hecke algebra associatedwith the orbit of a two sided Weyl group action on the Renner monoid of afinite monoid of Lie type, $M$. It is shown by Putcha in \cite{Putcha97} thatthe Kazhdan-Lusztig involution (\cite{KL79}) can be extended to the orbit Heckealgebra which enables one to define the $R$-polynomials of the intervalscontained in a given orbit. Using the $R$-polynomials, we calculate theM\"obius function of the Bruhat-Chevalley ordering on the orbits. Furthermore,we provide a necessary condition for an interval contained in a given orbit tobe isomorphic to an interval in some Weyl group.

R-POLYNOMIALS OF FINITE MONOIDS OF LIE TYPE