By studying two unsharp quantum structures, namely extended lattice ordered effect algebras and lattice ordered QMV algebras, we obtain some characteristic theorems of MV algebras. We go on to discuss automata theory based on these two unsharp quantum structures. In particular, we prove that an extended lattice ordered effect algebra (or a lattice ordered QMV algebra) is an MV algebra if and on...

The $q$-Onsager algebra $\mathcal O_q$ is defined by two generators $W_0, W_1$ and relations called the $q$-Dolan/Grady relations. Recently Baseilhac Kolb obtained a PBW basis for with elements denoted $\lbrace B_{n \delta+ \alpha_0} \rbrace_{n=0}^\infty, \lbrace \alpha_1} \delta} \rbrace_{n=1}^\infty $. In their recent study of current A_q$, Belliard conjecture that there exist W_{-k}\rbrace_{...