Let M be a symmetric Minkowski plane. By [1,2,6] there is a commutative field F such that the circles of M can be identified with the elements of the projective linear group P GL(2, F ). We consider only the caseWe introduce in M an order structure by a valuation on the set Λ of circles : "[ ] : Λ → {1, −1}" satisfying the condition :We show that there is a one to one correspondence with these order structures of M and the halforders of the field F . In a forthcoming paper the notion of "separation" for quadruples of concyclic points will be derived from a valuation and so the connection with H.-J. Kroll's (cf.[9]) concept of an ordered symmetric Minkowski plane established.
We give a description of automorphisms of symmetric and double symmetric chain structures. We use our results for double symmetric 1,2,3-structures to shed some new light on their groups of automorphisms.
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