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If G is countable then just enumerate the system Li and order it correspondingly. Let the elements of L~ precede those of L2. l,~ of left coset representatives (where 1 ~ lj ¯ L~ ULuand l~+~ it Li if/j ¯ Li), first by length and second ! lexigraphically. lm+l ! l,,+~. 1,~_~l~ (where l~ ¯ Li if lm¯ Li. Thus without loss of generality we assume that G is countable. Wedefine an ordering on the products of right coset representatives in the L~-1 by taking inverses. Wenow extend this ordering to the set of pairs {g,g-1},g ¯ G, where the notation is chosen so that the leading half of g precedes that of g-~ with respect to ordering _<.

Wenow extend this ordering to the set of pairs {g,g-1},g ¯ G, where the notation is chosen so that the leading half of g precedes that of g-~ with respect to ordering _<. Then we set {g,g-1} _< {h,h-~} if either L(g) < L(h) or L(g) = L(h) and the leading half of g strictly precedes that of h, or L(g) = L(h), the leading halves of -~. and h coincide, and the leading half of g-1 precedes that of h Thus if {g,g -1 } < {h,h -~} and {h,h -1} < {g,g-~} then g and h differ only in the kernel; since this can occur with g ¢ h then < is only a pre-order.

Further if instead M is an asynchronous auto~naton then (L, M) is an asyncronous automatic structure. A group G is an automatic group if it has an automatic structure. Similarly it is a biautomatic group if it has a biautomatic structure and an asynchronouslY automatic group if it has an asynchronous automatic structure. Biautomatic groups are clearly automatic but it is not knownwhether the two classes coincide. Further the class of asynchronously automatic groups is muchlarger than the class of automatic groups.