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This is what we study in the next section. 7. M o n o t o n e functions and semigroups A left ordered group (G, <_) is a group G endowed with a quasiorder < which satisfies x <_y:=~gx~gy Vg, x, y E G . Then S :-- {g E G: 1 _~ g} is a submonoid of G and if, conversely, S is a submonoid of G, then the assignment g ~_s g~ :¢::::~g~ E gS 30 1. Lie semigroups and their tangent wedges defines a left invariant quasiorder on G such that S := {g E G : 1 _

24), and therefore F := L(H) C_ker f ' ( 1 ) . Hence F M W C_ ker f'(1) M W C_ H ( W ) because f ' ( 1 ) E algint W*. 4). To see that Y is a Lie wedge, we have to show that Ad(H)V = V. This follows from the fact that (exp H ( W ) ) is dense in H , the closedness of V, and ead H(W)v C_ ead H(W)F + eadH(W)w (~_F + W = V. So V is a Lie wedge, f ' ( 1 ) E algint V*, and the analytic subgroup (exp H ( V ) ) = H is closed in G. 22 entails that V is global in G. 34. 36. (Characterization of the global Lie wedges) Let G be a connected Lie group and W C L(G) a Lie wedge.

Y and W t = ~ . X - ~ . (~ + ~ ) . (~ ~ + ~ ) ) . z . If S is a semigroup containing a whole neighborhood of Z0, this calculation shows that for a suitable choice of n, (, and 7/, the element U lies both in int S and in the XY-plane. Rotating W and W ' around the Z-axis we also find - U E int S. But then 1 = U * - U E i n t S , so that S = G. 2. The groups S1(2, 11%) and PSI(2, IR) 49 Note that taking the inverse image of a wedge in IR2 = G / Z ( G ) in G we find plenty of subsemigroups in G. In fact, for each wedge W in L(G) whose edge contains the center of L(G) gives us a semigroup S with L(S) = W .

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