7 problems

Let $\mathcal{A}$ and $\mathcal{B}$ be automata, and let $\Sigma(\mathcal{A})$ and $\Sigma(\mathcal{B})$ be the automaton semigroups generated by their states. Their free product i…

An automaton semigroup is a semigroup generated by the states of a finite synchronous automaton, and the free product of semigroups $S$ and $T$ is denoted by $S\star T$. The automa…

Let $S$ and $T$ be finite semigroups, and let $S\star T$ denote their free product. An automaton semigroup is a semigroup generated by the states of a finite synchronous automaton…

Let $F_2=\langle x,y\rangle$ be the free semigroup on $x$ and $y$, let $C_2=\{e,f\}$, and let $\phi:F_2\to C_2$ be the homomorphism defined by $\phi(x)=e$ and $\phi(y)=f$. Consider…

Let $S$ and $T$ be finite semigroups, and let $S\star T$ denote their free product. Free-product conjecture. There exist finite semigroups $S$ and $T$ such that $S\star T$ is not a…

Let a $p$-letter automaton generate a semigroup, let its growth series be the generating function recording the number of elements of each length, and let its dual be the automaton…

Let a reversible Mealy automaton be a Mealy automaton whose transition maps are permutations, and let the semigroup generated by an automaton be the semigroup generated by its stat…