On recognition and compactification
Let S be a semigroup and A a subset of S. The syntactic congruence of A is defined thus: \(x\sim_ Ay\) if and only if \(\{\) for all \(u,v\in S\), uxv\(\in A\) if and only if uyv\(\in A\}\). Recall that a necessary and sufficient condition for A to be recognizable in S is that the quotient semigroup \(S/\sim_ A\) be finite. A compactification of S is a compact topological semigroup T and a morphism of S onto a dense subsemigroup of T. For the Bohr compactification \(S\to \tilde S\), let \(\overset\circ S\) denote the image of S in the zero dimensional semigroup obtained by collapsing each component of \(\tilde S\) to a point. When there are enough morphisms of S to finite semigroups to separate points, S is called residually finite. The author obtains a variety of results about the syntactic congruence of A and various compactifications of S. One result is: if S is residually finite then for each \(A\subset S\), A is recognizable in S if and only if cl(A) is open in \(\overset\circ S\). A version of this for \({\mathcal P}\)- recognizable sets is also proved, where \({\mathcal P}\) is any pseudovariety of finite semigroups. An example is given of a group G and subset A which is recognizable in G because \(G/\sim_ A\) is finite, but \(G\to G/\sim_ A\) is not continuous and the image of A is not closed in G. When V is an alphabet of q letters, \(V_ q^{\infty}\) denotes the compact zero dimensional monoid consisting of the free monoid on V together with all infinitely long words on V. It is proved that \(V_ q^{\infty}\setminus \{1\}\) can be obtained as a compactification with respect to a certain pseudovariety.
scientific article; zbMATH DE number 4120428