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Partially ordered finite monoids and a theorem of I. Simon. (English) Zbl 0658.20035
A semigroup S is called $${\mathcal J}$$-trivial if Green’s relation $${\mathcal J}$$ is the identity on S, and a monoid M is called partially ordered (p.o.) if M admits a partial order $$\leq$$ such that the identity of M is the maximum element of M and such that ac$$\leq bd$$ in M whenever $$a\leq b$$ and $$c\leq d$$. It is immediate that every p.o. monoid is $${\mathcal J}$$-trivial, but that the converse is false. In this paper it is shown however that every finite $${\mathcal J}$$-trivial monoid is a homomorphic image of a finite p.o. monoid. The proof of this beautiful result is based on the ideal structure of finite $${\mathcal J}$$-trivial monoids and owes much to the theory of semigroup extensions studied by J. Birget and J. Rhodes [J. Pure Appl. Algebra 32, 239-287 (1984; Zbl 0546.20055)]. Related results on finite $${\mathcal J}$$-trivial monoids were found by H. Straubing [Semigroup Forum 19, 107-110 (1980; Zbl 0435.20036)]. As a result of this theorem a completely new proof of the theorem of I. Simon [Lect. Notes Comput. Sci. 33, 214-222 (1975; Zbl 0316.68034)] is obtained which characterizes those recognizable languages whose syntactic monoids are $${\mathcal J}$$-trivial (namely, the piecewise testable ones).
Reviewer: H.Mitsch

##### MSC:
 20M10 General structure theory for semigroups 68Q45 Formal languages and automata 06F05 Ordered semigroups and monoids 20M30 Representation of semigroups; actions of semigroups on sets
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##### References:
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