Perfect congruences on a free monoid

Abstract

Perfect congruences on a free monoid X* are characterized in terms of congruences generated by partitions of XU {1}. It is established that the upper semilattice of perfect congruences if V-isomorphic to the upper semilattice of partitions on XU {1}. A sublattice of the upper semilattice of perfect congruences is proved to be lattice isomorphic to the lattice of partitions on X. Introduction and summary. Free monoids derive their importance from the theory of formal languages. Their homomorphic images constitute the class of all monoids, so that their congruences give rise to isomorphic copies of all monoids. In view of the richness of a free monoid, any attempt at a reasonable classification of its congruences appears to be a daunting, not to say impossible, task. The classification of a very restricted type of congruence with some information on how these fit in the lattice of all congruences may, however, be possible. A congruence p on a monoid M is said to be perfect if the product of any two p-classes as complexes is a full p-class. (For related material see [2, VII 5.21 and VII 5.24].) In this paper we investigate the family PC(X*) of perfect congruences on a free monoid X* on the arbitrary alphabet X with three main goals in view, viz., to give an explicit description of such congruences; to determine the position of PC(X*) within the lattice C(X*) of all congruences on X*; and to establish properties of the partially ordered set PC(X*). We show, in particular, Theorem A, that p is perfect if and only if p is the congruence generated by the restriction of p to X U {1}. This characterization leads naturally to Theorem C wherein we prove that PC(X*) is a complete V-subsemilattice of C(X*) which is V-isomorphic to the complete V-semilattice II(X U {1}) of all partitions of X U {1}. Throughout the paper, for any congruence p on X* and any subset T of X*, PIT will denote the restriction of p to T and T* will denote the monoid generated by T. If ir is an equivalence relation on T, then lr* will denote the least congruence containing ir and uir the ir-class of u. The difference of two sets A and B will be denoted by A\B. As a general reference we recommend G. Lallement's book [1]. We start by proving a sequence of seven lemmas thus setting the stage for the proof of Theorem A. It will be convenient to introduce the following: Notation. For any A C X* and w E X*, let WA be the word obtained from w by deleting all letters from A which occur in w. Received by the editors December 2, 1985. 1980 Mathematics Subject Caasifiation (1985 Revision). Primary 20M05.