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On recognition and compactification. (English) Zbl 0684.20057

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.
Reviewer: J.M.Day

MSC:

20M35 Semigroups in automata theory, linguistics, etc.
20M07 Varieties and pseudovarieties of semigroups
22A15 Structure of topological semigroups
20M15 Mappings of semigroups
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References:

[1] Anderson, L. W. and R. P. Hunter,On the infinite subsemigroups of a compact semigroup Fund. Math. LXXIV (1972), 1–19. · Zbl 0231.22003
[2] Boasson, L. and M. Nivat,Adherences of languages J. of Com. and System Sciences 20 (1980), 285–309. · Zbl 0471.68052 · doi:10.1016/0022-0000(80)90010-0
[3] Hall, M.,A topology for free groups and related groups Ann. Math. 52 (1950), 127–139. · Zbl 0045.31204 · doi:10.2307/1969513
[4] Hunter, R.P.,On infinite words and dimension raising homomorphisms Fund. Math. (to appear). · Zbl 0663.22003
[5] Hunter, R. P.,Certain finitely generated compact zero dimensional semigroups J. Australian Math. Soc. (Series A) 44 (1988), 265–270. · Zbl 0649.22002 · doi:10.1017/S1446788700029852
[6] Lallement, G.,Semigroups and Combinatorial Applications, Wiley and Sons (1979). · Zbl 0421.20025
[7] Reutenauer C.,Une Topologie du monoide libre Semigroup Forum 18 (1979), 33–49. · Zbl 0444.68076 · doi:10.1007/BF02574174
[8] Schuppar, B.,Elementare aussagen über Funktionenkörper J. Reine Angew. Math. 313 (1980), 59–71. · Zbl 0423.12015 · doi:10.1515/crll.1980.313.59
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