Lie groups vs Lie monoids 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?
 A: See the following:


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*MR1317811  Hilgert, Joachim; Neeb, Karl-Hermann Lie semigroups and their applications. Lecture Notes in Mathematics, 1552. Springer-Verlag, Berlin, 1993. xii+315 pp. ISBN: 3-540-56954-5 (Reviewer: Gestur Ólafsson)

*MR1179336  Neeb, Karl-Hermann On the foundations of Lie semigroups. J. Reine Angew. Math. 431 (1992), 165–189. (Reviewer: Jimmie D. Lawson)

*MR1235759 Mittenhuber, Dirk; Neeb, Karl-Hermann Remarks on our paper: "On the exponential function of an invariant Lie semigroup'' [Sem. Sophus Lie 2 (1992), no. 1, 21–30; MR1188629 (93j:22007)]. Sem. Sophus Lie 3 (1993), no. 1, 119–120.
Moreover, completions of infinite dimensional Lie groups (like diffeomorphism groups) with respect to right invariant Riemannian metrics tend to be semigroups, for an easy example see 4.8 of here.
A: Anders Kock mentions Lie monoids and some of their properties in his book on synthetic differential geometry. Basically, in SDG, a Lie monoid is a microlinear monoid object. It is easy to show that the tangent space at the identity of a Lie monoid is an $R$-Lie algebra, defined in the same way as for a Lie group, and isomorphic to the left- or right-invariant vector fields on the monoid (depending on your bracket convention). We also still have a Lie functor, taking Lie monoids to their Lie algebras, and Lie monoid homomorphisms to Lie algebra homomorphisms.
For example, for any microlinear space $M$ the mapping space $M^M$ is a Lie monoid, with the composition of maps as the multiplication. Then the Lie algebra of $M^M$ is the space $\frak{X}$$(M)$ of vector fields on $M$. Since each infinitesimal transformation is invertible, it follows that this is also the Lie algebra of $\text{Diff}(M)$.
One thing to note is that since we do not have an inversion map on a Lie monoid, then I do not see how would have a canonical isomorphism of Lie algebras between left- and right-invariant vector fields on the monoid. Usually the isomorphism would be given by pushing forward a left- or right-invariant vector field by the inversion map.
A: 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. 
A: The basic theory is elaborated in Hilgert, Hofmann & Lawson's Lie Groups, Convex Semicones and Semigroups.
The analagous structure to a Lie algebra is what they call a Lie wedge. This being a wedge (a closed convex cone) $W$ that is invariant under the ad action. That is:

$e^{ad(v)}.W = W$ for all $v \in HW$.

Here $HW$ is the edge of the wedge and defined as $W \cap -W$. Lawson also defines subtangent vectors and these form a wedge and the tangent wedge to any local semigroup is a Lie wedge. It turns out that the converse is true but takes quite a bit of work. Two easy special cases are for strictly positive wedges and split wedges which are always Lie wedges.
Hilgert & Neeb's Lie Semigroups and applications focuses on a certain class of Lie semigroups, those that are a closed subsemigroup of a Lie group generated by the images of all one parameter semigroups lying in the ambient subsemigroup.
They choose this definition, they say:

"to avoid the complications arising from the semigroup analogues of dense winds on tori".

Presumably irrational tori, though I think I prefer their more colourful language.
You may also be interested in a different weakening of a Lie group. Loops are groups where the associative law has been weakened. The analogue to a Lie algebra for a Lie-Moufang loop is a Lie-Moufang or Mal'cev algebra. This also is an anti-symmetric non-associative algebra satisfying:

$(uv)(uw) = ((uv)w)u + ((vw)u)u + ((wu)u)v$

Not quite as symmetrical as the Jacobi law, but there you go. Both $\mathbb{O}^{\times}$ and $S(\mathbb{O})$,  the set of invertible and unit octonions respectively, form Moufang loops, so they turn up naturally. Notably octonions turn up in classifying supergravities coupled to super Yang-Mills and also in defining the Rosenfield projective planes which give a systematic geometric method in constructing the exceptional Lie groups upto a fully determined interpretation of 'systematic'.
A: In "Monoidal functors, species and Hopf algebras", by M. Aguiar and S. Mahajan (CRM Monograph Series, 29), they defined a Lie monoid (definition 1.25) in the following sense:
Let $(\mathsf{C}, \bullet, \beta)$ be a $\mathbb{K}$-linear symmetric monoidal category (possibly without unit). A Lie monoid in $(\mathsf{C}, \bullet, \beta)$ is a pair $(L, \gamma)$, where
$$\gamma: L \bullet L \to L$$
satisfies
$$\gamma+ \gamma \circ \beta_{L,L}=0 \qquad \text{and} \qquad \gamma \circ (\gamma \bullet \text{id})\circ (\text{id}+\xi + \xi^2)=0,$$
where $\xi$ denotes the composite
$$L \bullet L \bullet L\xrightarrow{\ \text{id}\,\bullet\,\beta_{L,L} \ }L\bullet L \bullet L \xrightarrow{\ \beta_{L,L}\,\bullet \,\text{id}\ }L \bullet L \bullet L.$$
Here "$\mathbb{K}$-linear" means that each set $\text{Hom}(A, B)$ of morphism in $\mathsf{C}$ is a $\mathbb{K}$-module and composition of arrows
$$\text{Hom}(A,B)\times \text{Hom}(B,C)\to \text{Hom}(A,C)$$
is $\mathbb{K}$-bilinear. The application $\beta$ is a natural isomorphism where for each $A, B \in \text{Obj}(\mathsf{C})$, the isomorphism
$$\beta_{A,B}: A \bullet B \to B \bullet A$$
satisfies $\beta_{A,B}\circ \beta_{B,A}=\text{id}$ and two axioms of compatibility with $\bullet$ (see definition 1.2).
