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Equations on the semidirect product of a finite semilattice by a $$J$$- trivial monoid of height $$k$$. (English) Zbl 0833.68073
Summary: Let $${\mathbf J}_k$$ denote the $$k$$th level of Simon’s hierarchy of $${\mathcal J}$$-trivial monoids. The 1st level $${\mathbf J}_1$$ is the $${\mathbf M}$$-variety of finite semilattice. In this paper, we give a complete sequence of equations for the product $${\mathbf J}_1 * {\mathbf J}_k$$ generated by all semidirect products of the form $$M* N$$ with $$M\in {\mathbf J}_1$$ and $$N\in {\mathbf J}_k$$. Results of Almeida imply that this sequence of equations is complete for the product $${\mathbf J}_1^{k+ 1}$$ or $${\mathbf J}_1*\cdots* {\mathbf J}_1$$ ($$k+ 1$$ times) generated by all semidirect products of $$k+ 1$$ finite semilattice and that $${\mathbf J}_1* {\mathbf J}_k$$ is defined by a finite sequence of equations if and only if $$k= 1$$. The equality $${\mathbf J}_1* {\mathbf J}_k= {\mathbf J}^{k+ 1}_1$$ implies that a conjecture of Pin concerning tree hierarchies of $${\mathbf M}$$-varieties is false.

##### MSC:
 68Q45 Formal languages and automata
##### Keywords:
trivial monoids; Simon’s hierarchy
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