By Greg Stewart

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**Example text**

Matveev 86, 87] Let K = (K, +, -( ), OK, " lK) be a ring. An algebra M = (M, (Wt)tEK) equipped with binary operations Wt: (x, y) f---+ txY (x EM, t E K) is said to be a geodetic algebra (algebra with geodesics) if ux(txy)=(ut)xY, l x Y=Y, (txy)=(l-t)yx (t, u E K, x, Y E M). 29) In this case the subsets (K)a b are called geodesics. 40. Remark. The notion of geodetic space (algebra) is a generalization of G-spaces [H. V. 41. Remark. Everything presented above may be considered if the operations are partially defined.

Q2 , idQ) then (Q, \/ , (e , id Q)) is a loop isomorphic to (Q, " e), and Q -+ (Q, id Q) = q t-+ (q, idq), is an isomorphism. Proof. Evident. 13. Proposition. V. Sabinin 72a,b] Let Q = (Q, " e) be a left loop and 1t be its transassociant. Then the canonical action of Q x 1t on Q, (q, h) X ~f (L q 0 h) x = q. hx, is faithful and transitive on Q, that is, under this action Q is a homogeneous space. In addition, i (H) = ((e, H), . , ( )-1, (e, idH)) is the stationary subgroup of the point e under the above action.

2) Q n [(q,h) . i(H)] = (Q,idQ) n [(q,h) . i(H)] = (Q,id Q) n (q,H) = {(q,id Q)}, thus Q is a cross section. If, further, q = 10 then Qn i(H) {(e, idQ)}. Finally, (q, id Q) . (10, L;1 ° h) = (q, h) implies Q . i (H) = Q x H. • SEMIDIRECT PRODUCTS 41 Proposition. V. Sabinin 72a,b] Let Q be a left loop and let pr Q : Q x H -+ (Q, id Q) = Q be defined by the rule pr Q (q , h) = (q, idq) . 12. q1 \/ [12 = (q1 ,idQ) \/ (q2 , id Q) = pr Q [ (q1 ,idQ) . (q2' idQ) ] = (q1 . q2 , idQ) then (Q, \/ , (e , id Q)) is a loop isomorphic to (Q, " e), and Q -+ (Q, id Q) = q t-+ (q, idq), is an isomorphism.