74

FIXED POINTS

WALTER TAYLOR

OF ENDOMORPHISMS

The theorem below gives necessary and sufficient conditions for all algebras 9/ in a variety r to have the following f ixed point property: i f f :9 /~9 .~ is any endomor- phism of 9/, then there exists aEA (the universe of 9I) such that f ( a ) = a . Namely, we prove that all algebras in r have the fixed point property if and only if r has a definable constant, i.e. there exists a term (i.e. polynomial symbol) ~ with one free variable in the language of ~ such that the equation a (Xo) -'- a (xl) holds identically in ~e'.

Our proof follows the general line 1) of Mal'rev [8], who gave necessary and suffi- cient conditions that all algebras in r have permutable congruence relations; his condition also was the existence of terms in the language of r satisfying certain identities relative to r Similar ('Mal'dev-type') conditions 2) have been adduced for joint distributivity and permutabitity of congruences (Pixley [9]), distributivity of congruences (J6nsson [7]), modularity of congruences (Day [3]), regularity of con- gruences (Cs~ik~iny [2]; Gr~itzer [6]; Wille [11]) and weak regularity of congruences Gr~itzer [6]). (Also consult Cohn [1, p. 148] or Gr~tzer [5, Theorem 26.4 and Exercises 5.69 and 5.70].)

We say that the algebra 9 /has the agreement property iff for every two endo- morphisms f and g of 9/ there exists aeA such that f (a )=g(a) . This property is strictly stronger than the fixed point property, as is seen in the remarks after the following theorem.

THEOREM. Let r be a variety of algebras defined by the set I of equations. Then the following conditions are equivalent:

(i) every algebra in r has the f ixed point property; (ii) the C/'-free algebra on No generators has the f ixed point property;

(iii) every algebra in r has the agreement property; (iv) the C/'-free algebra on two generators has the agreement property; (v) there exists a term ~ in the language of r having one free variable, such that

Z ~ ~ (Xo) ~--~ (xl) .

Proof. Clearly we need only show ( i i )~ (v) and (iv)=~ (v). First, to see (ii)=~ (v), let f be the unique endomorphism of the C/'-free algebra 9 / o n generators x~(ie~o)

i) But this may be the first instance of a MaI'rev-type theorem which can be proved only by considering the free algebra on an infinite set of generators; of. the remarks following the Theorem.

3) In fact~ c~diti~n ( v) ~f th~ The~rem is a str~ng Ma~ ~ev-type ~nditi ~n (see [ 6~ f~tn~te 2 ~).

Presented by G. Griitzer. Received January 25, 1972. Accepted for publication in final form Webruary 23, 1972.

FIXED POINTS OF ENDOMORPHISMS 75

satisfying the cond i t ion f (xi) = x~+ t (ieog). By (ii) there exists a ~A such that f ( a ) = a. There exists a term fl such that, for some k, a=fl(Xo .. . . , Xk_l) in 9~. Then fl(Xo . . . . , X k - 1 ) = a = f k ( a ) = f l ( X k . . . . , X2~-I) in 9/. Thus by substitution 27~-e(Xo~ " -~ (xl), where ~(Xo)is fl(Xo, Xo, ..., Xol.

Finally, to see (iv) =~ (v), let 9 /be ~r-free on generators x o and x 1, and let f u n d g be the unique endomorphisms ofg~ satisfying f ( x o) = f ( x l ) = Xo and g (Xo) = g (xl) = xl. Thus by (iv), there exists a e A such t ha t f ( a )=g(a ) . There exists a term fl such that a=fl(Xo, Xl) in 9~. Then fl(Xo, X o ) = f ( a ) = g ( a ) = f l ( x i , x~) ing~. And so 27 b a (Xo)-"- -"-a(xl) , where a(Xo)is fl(Xo, Xo). Q.E.D.

We note that condition (ii) of the Theorem is not equivalent to the condition that every finitely generated free algebra in ~e- have the fixed point property. Consider the variety of distributive lattices. Every free distributive lattice on a finite set of gener- ators is finite and hence a complete lattice; every lattice endomorphism is monotone, and according to a theorem of Tarski [10], every monotone function from a complete lattice to itself has a fixed point. Thus every finitely generated free distributive lattice has the fixed point property, but the variety of distributive lattices does not satisfy the equivalent conditions of the Theorem for (iii) clearly fails for 9.I taken as a two- element lattice.

Notice also that the variety of distributive lattices satisfies the condition formed from (iv) by replacing the free algebra on two generators by the free algebra on one generator (as does any variety of idempotent algebras).

Although one may prove some obvious generalizations of the Theorem above, we omit these because of the limitation which we now describe. For example, one may easily prove, for varieties ~e- having a distinguished binary operation (denoted by juxtaposition), that the equation x f ( x ) = x has a solution for all endomorphismsf of arbitrary 9.I e ~ if and only if there exists a term a such that

~ ' ~ c c , (*)

where ~' is formed from ~ by simultaneously replacing each occurrence of x~ by xi+ 1. But in contrast to the situation of the Theorem, there does not exist any a priori upper bound to the number of variables required in this ct. For one may check that if ~/ '= HSP (((0,1} n, F>}, where a) F ( (a i, ..., an>, (b I . . . . , bn>)= (a l , bl . . . . , bn- 1>, then no equation ( .) holds in ~e- with ~ having fewer than n variables, although (.) holds in #" for ~ = x l (x2 ... (xn-lxn)...).

REFERENCES

[1 ] P. M. Cohn, Universal algebra (Harper and Row, New York 1965). [2] B. Cs~ik~iny, Characterization of regular varieties, Acta. Sci. Math. (Szeged) 31, 187-189 (1970).

3) These (and related) binary operations were described in [4, w

76 WALTER TAYLOR

[3] A. Day, A characterization o f modularity for congruence lattices o f algebras, Canad. Math. Bull. 12, 167-173 (1969).

[4] T. Evans, Products of points -some simple algebras and their identities, Amer. Math. Monthly 74, 362-372 (1967).

[5] G. Gr~itzer, Universal algebra (van Nostrand, Princeton 1968). [6] G. Grfitzer, Two Mal'dev-type theorems in universal algebra, J. Combinatorial Theory 8, 334-342

(1970). [7] B. J6nsson, Algebras whose congruence lattices are distributive, Math. Scand. 21, 110-121 (1967). [8] A. I. Mal'6ev, On the general theory of algebraic systems (in Russian), Mat. Sb. N. S. 35 (77),

3-20 (I 954). [9] A. F. Pixley, Distributivity and permutability of congruence relations in equational classes of

algebras, Proc. Amer. Math. Soc. 14, 105-109 (1963). [10] A. Tarski, A lattice-theoretic fixpoint theorem and its applications, Pacific J. Math. 5, 289-309

(1955). [11] R. Wille, Kongruenzklassengeometrien, Lecture notes in mathematics ~ 113 (Springer-Verlag,

Berlin 1970).

University o f Colorado

Boulder, Colorado

U.S.A.