The theory of linear algebraic monoids has been developed in the last few years by L. Renner and the author [21-361 as a generalization of the theory of linear algebraic groups [16,40]. In applications of the theory of linear algebraic groups, one is naturally led to consider groups with BNpairs (or Tits systems) 12, 4, 9, 42,431. This class of groups includes in particular the groups of Lie type, so that the abstract properties of Tits systems are useful in studying the conjugacy classes and representations of groups of Lie type (see [4,9]). Our interest is in monoids. So let M be a monoid such that the group of units G has a BN-pair. In general G has very little to do with the rest of M. In fact, for a given G, M\G could be an arbitrary semigroup. So we introduce three axioms which intimately connect the idempotent structure of M with the structure of G. We then say that M is a monoid on G. The prime example is a connected regular linear algebraic monoid with zero. But there are other natural examples. For instance, Renner [37] is developing a theory of “monoids of Lie type” along the lines of Steinberg [41] for groups. Another example arises from the work of Smith [39] and Cabanes [3]. From their work, one can associate with certain irreducible modular representations, 4 : G + GL( V), a set of idempotents Es End(V). Then the monoid M generated by 4(G) and E is a monoid on 4(G) in the examples we have worked out, and very likely in general. In the situation of linear algebraic monoids, there is a natural type map from the finite lattice % of conjugacy classes of idempotents (g the lattice of f-classes) into the power set of the Dynkin diagram of G. The idea of the type map is due to Lex Renner and is exploited by Renner and the author [31] to show in particular that the system of idempotents can be constructed from the type map. In the present situation of monoids on G, there is again a type map I which characterizes the system of idempotents. 139 0021~8693/89 $3.00