f-disjunctive congruences and a generalization of monoids with length (Q915873)

An l-monoid is a monoid M endowed with a homomorphism \(\lambda\) into the natural numbers under addition such that \(0\lambda^{-1}=\{1\}\). If the condition that \(\lambda\) is a homomorphism is weakened requiring only that \((xy)\lambda \geq x\lambda +y\lambda\) for all x,y\(\in M\), then M is said to be a ql-monoid. The author shows that finitely generated ql- monoids are precisely the quotients of finitely generated free monoids by congruences whose classes are finite and also the finitely generated monoids in which every nonidentity element has only a finite number of factorizations as a product of nonidentity elements. He obtains as a corollary that the notions of l-monoid and ql-monoid coincide for finitely generated commutative monoids. A class of finite codes over a finite alphabet (known as solid codes) may be used to construct examples of ql-monoids, showing, in particular, that a finitely generated ql- monoid may not be an l-monoid. The same technique is used to construct an infinite strictly ascending chain of congruences on a finitely generated free monoid such that the corresponding quotients are l-monoids.