Forest Algebras, ω-Algebras and A Canonical Form for Certain Relatively Free ω-Algebras

Forest algebras are defined for investigating languages of forests [ordered sequences] of unranked trees, where a node may have more than two [ordered] successors [9]. We show that several parameters on forests can be realized as forest algebra homomorphisms from the free forest algebra into algebras which retain the equational axioms of forest algebras. This includes the number of nodes, the number of connected parts, the set of labels of nodes, the depth, and the set of labels of roots of an element in the free forest algebra. We show that the horizontal monoid of a forest algebra is finite if and only if its vertical monoid is finite. By an example we show that the image of a forest algebra homomorphism may not be a forest algebra and also the pre-image of a forest subalgebra by a forest algebra homomorphism may not be a forest algebra. Bojańczyk and Walukiewicz in [9] defined the syntactic forest algebra over a forest language. We define a new version of syntactic congruence of a subset of the free forest algebra, not just a forest language, which is used in the proof of an analog of Hunter’s Lemma [23] in the third chapter. The new version of syntactic congruence is the natural extension of the syntactic congruence for monoids in case of forest algebras. We show that for an inverse zero action subset and a forest language which is the intersection of the inverse zero action subset with the horizontal monoid, the two versions of syntactic congruences coincide. Almeida in [2] established some results on metric semigroups. We adapted some of his results to the context of forest algebras. We define on the free forest algebra a pseudo-ultrametric associated with a pseudovariety of forest algebras. We show that the basic operations on the free forest algebra are uniformly continuous, this pseudo-ultrametric space is totally bounded, and its completion is a forest algebra. The difficult part is how to handle the faithfulness property of forest algebras. We show that in a metric forest algebra with uniformly continuous basic operations, its horizontal monoid is compact if and only if its vertical monoid is compact. We show that every forest algebra homomorphism from the free forest algebra into a finite forest algebra is uniformly continuous. We show that the analog of Hunter’s Lemma [23] holds for metric forest algebras, which leads to the result that zero-dimensional compact metric forest algebras are residually finite. We