An Algebraic Characterization of Prefix-Strict Languages - Jing Tian; Yizhi Chen; Hui Xu (Mathematics, 2022)
Let <semantics>Σ+</semantics> be the set of all finite words over a finite alphabet <semantics>Σ</semantics>. A word u is called a strict prefix of a word v, if u is a prefix of v and there is no other way to show that u is a subword of v. A language <semantics>L⊆Σ+</semantics> is said to be prefix-strict, if for any <semantics>u,v∈L</semantics>, u is a subword of v always implies that u is a strict prefix of v. Denote the class of all prefix-strict languages in <semantics>Σ+</semantics> by <semantics>P(Σ+)</semantics>. This paper characterizes <semantics>P(Σ+)</semantics> as a universe of a model of the free object for the ai-semiring variety satisfying the additional identities <semantics>x+yx≈x</semantics> and <semantics>x+yxz≈x</semantics>. Furthermore, the analogous results for so-called suffix-strict languages and infix-strict languages are introduced.