Out Subword-Free Languages and Its Subclasses

Abstract

Let [Formula: see text] be a finite alphabet and [Formula: see text] the set of all words over [Formula: see text]. A subword-free language (also known as a hypercode) is an independent subset of [Formula: see text] with respect to the embedding order (denoted by [Formula: see text]) on [Formula: see text]. In this paper we introduce three subsets of the partial order [Formula: see text], and study three subclasses of languages defined by these subsets. They are out subword-free languages, left subword-free languages and right subword-free languages. The properties of these languages are established for determining their combinatorial and algebraic structures. By equipping them with two binary operations, respectively, all these classes of languages form semilattice-ordered semigroups. It is shown that they are freely generated by [Formula: see text] in three subvarieties of semilattice-ordered semigroups, respectively. It is also shown that the word problems for these free algebras are solvable.