GRATZER, G. Math. Annalen 170,334-338 (1967)
On the Endomorphism Semigroup of Simple Algebras*
GEORGE GRATZER
* The preparation of this paper was supported by the National Science Foundation under grant GP-4221. i. Introduction. Statement of results Let (A; F) be an algebra andletE(A ;F) denotethe set ofall endomorphisms of(A ;F). Ifep, 1.p EE(A ;F)thenwe define ep •1.p (orep1.p) asusual by x(ep1.p) = (Xlp)1.p. Then (E(A; F); -> is a semigroup, called the endomorphism semigroup of (A; F). The identity mapping e is the identity element of (E(A; F); .). Theorem i. A semigroup (S; -> is isomorphic to the endomorphism semigroup of some algebra (A; F) if and only if (S; -> has an identity element. Theorem 1 was found in 1961 by the author and was semi~published in [4J which had a limited circulation in 1962. (The first published reference to this is in [5J.) A. G.WATERMAN found independently the same result and had it semi~ published in the lecture notes of G. BIRKHOFF on Lattice Theory in 1963. The first published proofofTheorem 1 was in [1 Jby M.ARMBRUST and J. SCHMIDT. The special case when (S; .) is a group was considered by G. BIRKHOFF in [2J and [3J. His second proof, in [3J, provides a proof ofTheorem 1; namely, we define the left~multiplicationsfa(x) == a . x as unary operations on Sand then the endomorphisms are the right~multiplicationsxepa = xa. We get as a corollary to Theorem 1BIRKHOFF'S result. Furthermore, noting that if a (ES) is right regular then fa is 1-1, we conclude Corollary 2. A semigroup (S; .) is isomorphic to the semigroup of all 1-1 endomorphisms of an algebra if, and only if, it has an identity element and it is right regular. The problem of characterizing the semigroups of 1-1 endomorphisms and onto endomorphisms within the endomorphism semigroup was solved by M. MAKKAI [5]. In 1961 the author observed that there exist simple algebras 1 with a given automorphism group and there exist algebraswith oneelement automorphism group and arbitrary congruence lattice. The question was raised whether the automorphismgroupis independentofthecongruencelattice. Apositiveanswer was given by E. T. SCHMIDT [6]. 1 Let A be a set; the binary relations ro and I on A are defined by the rules: x == y(ro) iff x "" y ; X ==Y(I) for all x, yE A. An algebra
The next question is the independence of the congruence lattice and of the endomorphism semigroup. A negative answer to this question is given by the following result. Theorem 3. A semigroup (S,.·) is isomorphic to the endomorphism semigroup of some simple algebra (A,. F) if and only if (S,. .) has an identity element and every element in S is either right regular or a right annihilator. Let us assume that
(b) if fn(r t ) = m, then fn(r t rz) = fm(r z). Theorem 4. Let Problem 4. Let Iff be a semigroup ofmappings ofa set A into itself, containing the identical mapping. Under what conditions is Iff the semigroup of all endo morphisms of an algebra 2. Proofofthe results To prove Theorem 3 let (A; F) be an algebra, IpEE(A; F). Let 8" be the relation on A defined by x == y(8,,) ifXlp = Yip. Then 8" is a congruence relation of Set F = Uala ES} V {n, p}. To show that Ipa for a E R: Xlpa=xa if XES; Xlpa=X ifx¢S. Ipa for aEN :Xlpa=a, for all xEA. The Endomorphism Semigroup 337 It should be noted that on S these definitions coincide with the ones given in the sketched proof of Theorem 1. This implies: (i) If qJ EE(A ; F) and qJ maps S into S then qJ restricted to S coincides with one of the qJa restricted to S. (ii) If qJ EE(A; F) and qJ is 1-1, then OqJ = 0 and 1qJ = 1. Indeed, 1qJ = a =F 1 would imply xqJ=(xn1)qJ=xqJn1qJ=xqJna, thus xqJ=a or 0, a contra diction. (iii) If qJ EE(A; F) and qJ is 1-1, then qJ = qJa for some a ER. Let XES; then XqJ is in S; otherwise XqJ = OqJ or XqJ = 1qJ. Thus and (E(A; F); '). This completes the proof of Theorem 3. Theorem 4 can be proved by direct computation. To prove Theorem 6, let a E G, a =F e and define a binary relation eO. on A by the rule: x == y(e a.) if xaft = y for some integer n. Then eO. is an equivalence relation. Since every IE F is unary, it is also a congruence relation. (This does not hold in general!) Since a =F e, we have eO. =F w. Using that (A; F) is simple, we conclude that eO. = I. This means that A = {xaftln =0, ± 1, ±2, ...} for any fixed x EA. If a is not of prime order, then there exists a positive integer m, such that e =F P= am generates a subgroup G1 properly contained in the sub group G2 generated by a. Let us note that x = xy(y EG1) implies y = e. Indeed, if x = xy then x = xyft for all integers n, thus e v =F I. Therefore, y = e. If y, bE G 1, xy = xb, then x = X(by-l), thus by-l = e, i.e. y = b. Now let y EG2 , Y¢ G1• Then, by representing xy as xPft (what we can do since P=F e implies e p = I), we conclude that y = pft, which contradicts y ¢ G1• It follows from the previous remark that xaft = xamiff aft = am, thus if a =F e, 1 and a is of order p then x, xa, ..., xaP- are all distinct, thus A has p elements. This concludes the proof of Theorem 6. Remark (added in proof, Jan. 22, 1967): Problem 3 has been solved by W. A. LAMPE. The semigroups in question are the following: the one element group, groups ofprime order, the idem potent two element semigrou.p, with identity, and the full transformation semigroup on a two element set. 338 G. GRATZER: The Endomorphism Semigroup References [Il ARMBRUST, M., and J, SCHMIDT: Zum Cayleyschen Darstellungssatz, Math. Ann. 154,70-72 (1964), [2] BIRKHOl'F, G.: On the structure of abstract algebras. Proc. Cambridge Phil. Soc, 31, 433--454 (1935). [3] _. On groups of automorphisms, Rev. Un. Mat. Argentina t:t, 155-157 (1946). [4] GRATZER, G.: Some results on universal algebras. M. meographed notes, August 1962. [5] MAKKAI, M.: Solution of a problem of G. Griitzer concerning endomorphism semigroups. Acta Math. Hung, 15,297-307 (1964). [6] SCHMIDT, E. T.:UniversaleAlgebren mitgegebenenAutomorphismengruppenund Kongruenz- verbanden. Acta Math. Hung. 15, 37--45 (1964). Professor G. GRATZER Department of Mathematics Pennsylvania State University University Park, Pa. U.S,A, and Department of Mathematics and Astronomy The University of Manitoba Winnipeg, Man., Canada (Received July 13, 1965)