Test sets for finite substitutions

The Ehrenfeucht Conjecture [5] states that if Μ is a finitely generated free monoid with nonempty subset S, then there is a finite subset T⊂S (a “test set”) such that given two endomorphisms f and g on Μ, f and g agree on S if and only if they agree on T. In[4], the authors prove that the above conjecture is equivalent to the following conjecture: a system of equations in a finite number of unknowns in Μ is equivalent to a finite subsystem. Since a finitely generated free monoid embeds naturally into the free group with the same number of generators, it is natural to ask whether a free group of finite rank has the above property on systems of equations. A restatement of the question motivates the following.

Hypercodes

We prove that the test rank of a free metabelian Lie algebra L of rank n equals n-1 and a test set of L belongs to the derived subalgebra of L.

Lyndon words have been extensively studied in different contexts of free Lie algebra and combinatorics. We introduce Lyndon partial words, arrays and trees. We also study free monoid morphisms that preserve finite Lyndon partial words and check whether a morphism preserves or does not preserve the lexicographic order. We propose an algorithm to determine Lyndon partial words of given length over the binary alphabet. Image analysis in several way of scanning via automata and grammars has a significance in two-dimensional models, we connect 2D Lyndon partial words with few automata and grammar models.KeywordsLyndon partial words2D Lyndon words2D arraysMathematics Subject Classification:68Q4568Q7068R1568U10

Given a (finite or infinite) subset X of the free monoid \(A^*\) over a finite alphabet A, the rank of X is the minimal cardinality of a set F such that \(X \subseteq F^*\). A submonoid M generated by k elements of \(A^*\) is k-maximal if there does not exist another submonoid generated by at most k words containing M. We call a set \(X \subseteq A^*\) primitive if it is the basis of a |X|-maximal submonoid. This extends the notion of primitive word: indeed, \(\{w\}\) is a primitive set if and only if w is a primitive word. By definition, for any set X, there exists a primitive set Y such that \(X \subseteq Y^*\). The set Y is therefore called a primitive root of X. As a main result, we prove that if a set has rank 2, then it has a unique primitive root. This result cannot be extended to sets of rank larger than 2.

We consider pure Lindenmayer systems, more precisely, 0L and T0L systems as language accepting devices and compare them to their generating counterparts. Accepting Lindenmayer systems can be seen as systems of inverse finite substitutions which are iteratively applied over a free monoid. Hereby, we investigate the deterministic case in detail, comparing several different concepts of determinism in such systems. Whereas in the usual generating case these concepts trivially are equally powerful, the structure of families of accepted languages is much richer. In passing, the case of unary Lindenmayer systems is investigated.

Algebraic effects and handlers are a convenient method for structuring monadic effects with primitive effectful operations and separating the syntax from the interpretation of these operations. However, the scope of conventional handlers are somewhat limited as not all side effects are monadic in nature.This paper generalizes the notion of algebraic effects and handlers from monads to generalized monoids, which notably covers applicative functors and arrows. For this purpose we switch the category theoretical basis from free algebras to free monoids. In addition, we show how lax monoidal functors enable the reuse of handlers and programs across different computation classes, for example handling applicative computations with monadic handlers.