Equations and dot-depth one.(English)Zbl 0814.20048

Let $$V_{k,m}$$ denote the finite monoid varieties corresponding to the Straubing hierarchy of star-free languages. The author studies whether these varieties are finitely based. For every pair of integers $$r \geq 0$$, $$m \geq 1$$ a finite sequence of equations $$C_ m^ r$$ is constructed such that $$C_ m^ 0 = \{x^ m = x^{m+1}\}$$ and $$C_ m^ r \sqsubseteq C_ m^ s$$ if $$r \leq s$$.
Theorem 1: Let $$M$$ be a monoid generated by a set of $$r + 1$$ elements, $$r \geq 0$$. Then $$M$$ belongs to $$V_{1,m}$$ iff $$M$$ satisfies the equations in $$C_ m^ r$$. The equations $$\bigcup\{C_ m^ r : r \geq 0\}$$ define the variety $$V_{1,m}$$. Theorem 2: The family of equations $$\bigcup\{C_ m^ r : r \geq 0\}$$ is equivalent to $$C_ m^ s$$, where $$s= 1$$ if $$m \in \{1,2\}$$ and $$s = 2$$ if $$m = 3$$. Thus the variety $$V_{1,m}$$ is finitely based if $$m = 1,2,3$$. The author also constructs for every $$m$$ a finite sequence of equations $$D_ m$$ such that for every finite monoid $$M$$ generated by $$2$$ elements, $$M \in V_{2,1}$$ iff $$M$$ ultimately satisfies the equations in $$\bigcup\{D_ m : m \geq 1\}$$.

MSC:

 20M35 Semigroups in automata theory, linguistics, etc. 20M07 Varieties and pseudovarieties of semigroups 68Q45 Formal languages and automata 20M05 Free semigroups, generators and relations, word problems
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References:

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