Forest algebras were defined for investigating forests [ordered sequences] of unranked trees, where a node may have more than two [ordered] successors [2]. 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 are defined by morphisms from the free forest algebra into algebras which retain the equational axioms of forest algebras. We present finiteness conditions for forest algebras.

We present a new version of syntactic congruence of a subset of the free forest algebra, not just a forest language [2], which leads to more general results. For a special subset and a forest language which is the restriction of the special subset to the horizontal monoid, the two versions of syntactic congruences coincide.

We define on the free forest algebra a pseudo-ultrametric associated with a pseudovariety of forest algebras. 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. We show that the analog of Hunter's Lemma [3] holds for metric forest algebras, which leads to the result that zero-dimensional compact metric forest algebras are residually finite.

We define $\omega$-algebras, which retain the equational axioms of forest algebras and are endowed with additional unary operations. We present some useful properties of the free $\omega$-algebra which implies that it is a forest algebra. A profinite algebra is defined to be a projective limit of a projective system of finite algebras [1]. It is natural to study the free pro-BSS algebra as an $\omega$-algebra. We present some results for sets of additively irreducible summands, multiplicatively irreducible factors, and some others concerning the type of an element of the free $\omega$-algebra which lead to an algorithm to compute a canonical form for each element of the free $\omega$-algebra in a certain variety. This allows us to identify the structure of the free pro-BSS algebra.

References

[1] J. Almeida, Profinite semigroups and applications, Published in V. Kudryavtsev, I. G. Rosenberg (Eds.), Structural Theory of Automata, Semigroups, and Universal Algebra, Proceedings of NATO Advanced Study Institute, Montréal, Québec, Canada, 7-18/7/2003. NATO Science Series II: Mathematics, Physics and Chemistry, Vol. 207, Springer, 2005, 1-45. 

[2] M. Bojanczyk and I. Walukiewicz, Forest algebras, Logic and Automata, (2008) 107-132.

[3] R. P. Hunter, Certain finitely generated compact zero-dimensional semigroups, J. Austral. Math. Soc., Ser. A 44 (1988) 265-270.