Equations and dot-depth one
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 \(\bigcup\{D_ m : m \geq 1\}\).
scientific article; zbMATH DE number 438889
scientific article