Stephen L. Bloom

On the algebraic structure of rooted trees

Figure 1.1 Figure 1.2 These two digraphs, while different, usually represent the same phenomenon, say, the same “computational process.” Our interest in rooted trees stems from the fact that these two digraphs “unfold” into the SAME infinite tree. In some cases at least it is also true that different (i.e. non-isomorphic) trees represent different phenomena (of the same kind). In these cases the unfoldings (i.e. the trees) are surrogates for the phenomena.

Solutions of the Iteration Equation and Extensions of the Scalar Iteration Operation

We study the solutions to a (vector) equation somewhat analogous to the traditional equations of linear algebra. Whereas, in introductory linear algebra the domain of discourse is the field of real numbers (or an arbitrary field) our domain of discourse is the algebraic theory of (multi-rooted, leaf-labeled) trees (or, more generally, any iterative theory).As in linear algebra, we obtain a necessary and sufficient condition for our equations to have unique solutions and we can describe “parametrically” the totality of solutions. However, whereas in linear algebra, there is no way of giving

Free shuffle algebras in language varieties

1 \div 0

Varieties of ”If-Then-Else“

meaning in such a way that all the “old laws” hold, we can give meaning to the “iteration operation” (the analogue of division into 1) in such a way that all the “old laws” still hold. Indeed, we can describe “parametrically” all such ways of extending the (partially defined) scalar iteration operation to all trees (more generally, morphisms).

Long words: the theory of concatenation and o-power

We give simple concrete descriptions of the free algebras in the varieties generated by the “shuffle semirings” LΣ := (P(Σ∗),+,., ⊗, 0,1), or the semirings RΣ := (R(Σ∗),+,., ⊗,∗,0,1), where P(Σ∗) is the collection of all subsets of the free monoid Σ∗, and R(Σ∗) is the collection of all regular subsets. The operation x ⊗ y is the shuffle product.

The equational theory of regular words

Four classes of algebras are considered. The algebras in each class contain functions whose behavior models a version of the “if-then-else” instruction. In one version, for example, the algebras contain a function

Iteration theories of synchronization trees

\kappa

Axiomatizing schemes and their behaviors

of four arguments such that

Matrix and matricial iteration theories, Part II

\kappa (x,y,u,v) = u

Scalar and vector iteration

if