Does there exist a well developed theory of a class of objects which might rightfully be called Lie monoids? By this I mean with axioms similar to those of Lie groups, but with the axiomatic existence of inverses dropped. If so what is the analogous structure to lie algebras if one exists?



Lie groups vs Lie monoids
There is a well developed theory of algebraic monoids, due principally to Putcha and Renner. I think Lie semigroups is less well developed but there is work by Hoffmann, Lawson and the thesis of Langlands was on this subject. 


