Abstract
Mathematically, the simplest and most natural operation considered in language theory is a morphism between two free monoids. Many of the problems concerning such morphisms are difficult and challenging. This chapter discusses the problems concerning morphism between two free monoids in formal language theory. The active areas of research, that is, equality sets and grammar forms, fit well into the framework of studies dealing with morphisms. A daily oral language (DOL) system constitutes a very simple finitary device for language definition. Languages defined by a DOL system are referred to as DOL languages. The DOL systems constitute a convenient framework for certain properties of ω-words. The chapter discusses two problems: the construction of a strongly cube-free ω-word over an alphabet with cardinality two, that is, strong cube-freeness problem and the construction of a square-free ω-word over an alphabet with cardinality three, that is, square-freeness problem.
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