Studia Scientiarium Mathematicarum Hungarica 15. (1980)

1-3. szám - Sullivan R. P.: Automorphisms of injective transformation semigroups

R. P. SULLIVAN Theorem 1. If S is any non-zero @x-normal transformation semigroup then either S contains a constant or for all non-zero ad S, rank a= \X\, |T\dom a)-= |ЛГ| and if |T\Ta| = |Z| then |xa-1|<|Z| for ail x€rana. Proof. Let a be any non-zero element of S with minimal rank equal to r (finite or infinite); we first show that if there exists YQX with |T|=r and |T\Tj = = |T\ran a| then Y Odom a= □ or |Ta|—r. For, if Y exists then there also exists gd&x with (Xa)g= Y ; since gag~xdS we have: [Xcc-gag-1] = \Xaga\ S \Xa\ = r, and so rank (aga) equals 0 or r. But (Xag)a=Ya and thus we have the desired conclusion. Suppose both r and X are finite, and let y, zdXa, a$Xa. If dom cclTran a — □ and if xdya~x, then a(x, y)a(x, y)dS and is a constant; hence we may suppose y€dom a. In fact, if now z|dom a then a2 has non-zero rank less than r : so, we can assume Ta Ljdom a. If a$dom a, then a (a, z)a(a, z)dS and has non-zero rank less than r\ hence we may also assume a6dom a, in which case уа~г contains at least 2 elements for some ydXa. In this event we can easily choose Y ddX so that |T|=r, |Т\У|= |A"\ran a|, Tfidom a = T and \Ya\—r—\, contradicting our very first remark. Now suppose r<\X\ when Xis infinite. If |doma[<|T| then |TT\dom a| = |T| and, fixing a€doma, we can choose I'LL with YПdom a = {a}, \Y\=r and |TT\T| = |T\ran a\—\X\. Then by our earlier remark j Ta j ;= r where in fact Ya—{aa}. On the other hand, if ]doma| = |Z| then there exists bdran a with \ba~x\>r. This latter observation is clear when r is finite; for infinite r note that doma= = U {xa-1: x£ran a} and hence if |xa-1| ^r for all x£ran a then |dom a|Шг2=г, a contradiction. Now choose TLjèa-1 with |T|=r and |T\T| = |T\ran a| = — \X\: by a previous remark \Ya\~r where in fact Ya={b}. Hence in either case, a is a constant. If r=\X\ then every non-zero adS has ranka=|Z| and |doma| = |Tf|. If X is finite then S is a normal subgroup of dSx, possibly with □ adjoined. Suppose X is infinite and |Z\dom a| = |Tf | for some non-zero ad S. If in addition |Z\Ta| = = |Tf|, fix a£doma and let f=(L\doma)Ua. Then \Y\=r, |T\T| = |T\ran a| = = |Tf| and by an earlier remark |Ta|=r, clearly a contradiction. Hence |Tf\Ta|-< <|TT|. Now, if rank a2=0 then La(jdoma= □, contradicting the last statement; hence rank a2 =\X| and so |Zafldom a| = |Tf|. Let gd^x be the union of а 1—1 map from T\dom a onto dom a and one from dom a onto T\dom a and put ß=gag~x. Then BdS, ß^O, dom «LlXdom ß and dom a Lj X\Xß ■ that is, "lA’Xdom jß| = |Z| and |T\Zjß| = |T|, contradicting the first conclusion of this paragraph. Hence |T\dom a|<|Tf| for all non-zero a dS. Finally, suppose |Tf\Ta| = |X| for some non-zero a dS and let \ba~1\ = \X\ for some bdran a. Put Y=ba~x. Then |T|=r and |Z\T| = |T\ran a| — |Т|Г| and returning to the first paragraph we are forced to conclude that r— 1, a contradiction. Since for arbitrary X, the semigroup S contains a non-zero element with rank less than |Tf| or it does not, the proof is now complete. Before proceeding we note that if S is a non-zero -normal transformation semigroup defined on an infinite set X and □ d S but S does not contain a constant then □ ((»S2: that is, S can be regarded as a semigroup with □ adjoined. For, if Studia Scientiarum Mathematicarum Hungarica 15 (1Э80)