Varieties of automata and transformation semigroups (Q1207386)

For a prime \(p\), denote by \(C^ 1_ p=(p=\{0,1,\dots,p-1\},\{x_ 0,x\},\delta_ p)\), \(D_ 0=(2 =\{0,1\},\{x,y\},\delta_ 0)\), \(A_ 0=(2,\{x_ 0,x,y\},\delta_ 1)\) the automata such that \(\delta_ p(i,x_ 0)=i\), \(\delta_ p(i,x)=i+1\) where the addition is taken modulo \(p\), \(\delta_ 0(i,x) =\delta_ 1(i,x)=0\), \(\delta_ 0(i,y) =\delta_ 1(i,y)=1\), \(\delta_ 1(i,x_ 0)=i\). Let \(A_ i=(Q_ i,X_ i,\delta_ i)\), \(i=1,2,\dots,n\) be automata and \(\varphi_ i:\biggl(\prod^{i-1}_{j=1} Q_ j\biggr)\times X\to X^*_ i\), \(i=1,2,\dots,n\) be mappings, then the automaton \(A=(\prod^ n_{i=1} Q_ i,X,\delta)\) is an \(\alpha^*_ 0\)-product of \(A_ 1,A_ 2,\dots,A_ n\) if \(\delta((q_ 1,q_ 2,\dots,q_ n),x)=(\delta_ 1(q_ 1,x_ 1),\delta_ 2(q_ 2,x_ 2),\dots,\delta_ n(q_ n,x_ n))\) where \(x_ i =\varphi_ i(q_ 1,q_ 2,\dots,q_{i-1},x)\) for any \(i=1,2,\dots,n\). If \(\text{Im}(\varphi_ i)\subseteq X^ +_ i\) (or \(\text{Im}(\varphi_ i)\subseteq X_ i\cup\{\lambda\}\), or \(\text{Im}(\varphi_ i)\subseteq X_ i)\) for each \(i\) then we speak about an \(\alpha_ 0^ +\)-product (or an \(\alpha_ 0^ \lambda\)- product, or an \(\alpha_ 0\)-product, respectively). A class \(\mathcal V\) of automata is called an \(\alpha_ 0\)-variety (or an \(\alpha^ \lambda_ 0\)-variety, or an \(\alpha^ +_ 0\)-variety, or an \(\alpha^*_ 0\)- variety) if \({\mathcal V}\) is closed under \(\alpha_ 0\)-products (or \(\alpha^ \lambda_ 0\)-products, or \(\alpha^ +_ 0\)-products, or \(\alpha^*_ 0\)-products, respectively), subautomata, and quotient automata. The author proves that any \(\alpha_ 0\)-variety containing \(D_ 0\) and \(C^ 1_ p\) for any prime \(p\) is an \(\alpha^ +_ 0\)- variety, any \(\alpha_ 0\)-variety containing \(A_ 0\) and \(C^ 1_ p\) for any prime \(p\) is an \(\alpha^*_ 0\)-variety, any \(\alpha^ \lambda_ 0\)-variety containing \(A_ 0\) and \(C^ 1_ p\) for some prime \(p\) is an \(\alpha^*_ 0\)-variety. Some counterexamples are given.