On the congruence lattices of unary algebras

Abstract

Characterizations of those unary algebras whose congruence lattices are semimodular or atomic are obtained. Combining these results gives necessary and sufficient conditions for a unary algebra to have a geometric congruence lattice. A unary algebra, 91= (A; f ), is a set A together with a function f from A into A. A congruence relation 0 of 91 is an equivalence relation on A such that if x=y(O) then f (x)_f (y)(0). The congruence lattice of 9, denoted ?(91), is the collection of all congruence relations of W partially ordered in the usual manner [2, p. 50]. If L is a lattice and a, b E L, then b is said to cover a if a<b and {xla<x<b}= 0 . This relation will be denoted a-<b. L is semimodular if for any a, b and c E L such that c<a and c-<b then a<avb and b<avb. Theorem gives a characterization of those unary algebras whose congruence lattices are semimodular. An atom of a lattice with 0 is an element that covers 0. A lattice is atomic if every element is a join of atoms. Theorem 2 gives a characterization of those unary algebras whose congruence lattices are atomic. Let 9t =(A; f) be a unary algebra. For any x c A and any positive integer n letf0(x)=x andfn(x)=f(fn-l(x)). A component of W is a subset of A such that x, y are in the same component if and only if there exist m, n E N such that f I (x)-=f n(y). Index the components of 91 by an index set so that A = U ic Ai. For each component Ai, choose a fixed subset Ci, called a cycle, as follows: If there exists n c N and x & Ai such that fn(x)=x, let C1 = {fk(x)jk E N}. Such a cycle is a unique finite subset of A,. Denote such a finite cycle by {x0, x1, * x, i} with f(x)=xi+1. If n is any integer let xn=xi where n-i (mod m), 0<i<m. If no such x and n exist for Ai let x0 be any fixed element of Ai, Ci={fm(x0)|m E N}. Denote this set by {x0, x1, x2, * * } where f(xi)=xi+,. If C1 and Cj are any two cycles, ICil=mi and ICj1=mj, then C1 and Cj are said to have relatively prime cardinalities if both m1 and m3 are finite and (mj, mj)= or if mi is Received by the editors November 16, 1971. AMS 1970 subject classiflcations. Primary 08A25, 06A20; Secondary 05B35, 94A25.