Versatile monoids and versatile categories

Abstract

Let M be a monoid (i.e., a semigroup with identity), and let L be an algebraic lattice. The pair (M, L ) is representable if there exists a universal algebra whose endomorphism semigroup is isomorphic to M and whose subalgebra lattice is isomorphic to L. Such an algebra will be called a representation of (M, L) . The general problem of characterizing representable pairs is open, but a few partial results are known. For instance, Michael Stone [8] gives for each integer n>~2 an example of a monoid M such that (M, Cn + 1) is representable but (M, Cn} is not, where Cm denotes an m-element chain. A monoid M is called versatile if (M, L ) is representable for every algebraic lattice L having more than one element. Only algebraic (i.e. complete and compactly generated) lattices are considered, because, according to the well-known theorem of Birkhoff and Frink [1], only algebraic lattices can occur as subalgebra lattices. Furthermore, if an algebra has a one-element subalgebra lattice, then it is rigid (i.e. has a one-element endomorphism monoid), so we exclude this trivial case. In this paper we give a complete characterization of versatile monoids. A monoid M is versatile if and only if (M, C2) is representable, and this is equivalent to the condition that every element of M be either left cancellative or a left zero. The first of these results is a generalization of a theorem of E. T. Schmidt [7], and the proof makes use of some of Schmidt's techniques. The second result dualizes G. Gr/itzer's characterization of the endomorphism semigroups of simple algebras [3], and the proof uses some techniques of M. Makkai [5]. This result was also established independently by Stone [8] for the case of a finite monoid. In w 3, we define versatile categories and state similar results for small categories. Terminology and notations of universal algebra are largely those of [4]. Homomorphisms are written to the right of their arguments, while operations and polynomials are written to the left. I f 9~ = (A; F ) is an algebra, its subalgebra lattice and endomorphism monoid will be denoted by S(N) and E(9~), respectively. I f B ~ A , the carrier set of the subalgebra of 9~ generated by B will be denoted by [B]~. By convention, [0"]~ = 0 if F contains no nullary operations, and otherwise [0]~a is the set of all values of nullary polynomials.