One-dimensional tiling semigroups. (Q1434111)

A language \(L\) over a finite alphabet \(\Sigma\) is called factorial if any proper non-empty factor of any word from \(L\) belongs to \(L\). Inverse semigroups \(S(L)\) determined by factorial languages are investigated. Properties of \(S(L)\) and constructions of \(S(L)\) are proved. These semigroups have two important consequences. A tile is a connected closed bounded subset of \(\mathbb{R}^n\) such that it is equal to the closure of its interior. A sequence of tiles covering \(\mathbb{R}^n\) is called tiling if the intersection of two distinct tiles from this sequence is a subset of its boundaries. A tool for the study of tilings are semigroups determined by them. A tiling semigroup of a one-dimensional tiling \(\mathcal T\) is isomorphic to \(S(L)\) for the language \(L\) determined by \(\mathcal T\). The second application of inverse semigroups \(S(L)\) concerns semigroups describing shift spaces (a shift space is a set of bi-infinite words with forbidden factors).