We show that the assignment of the (left) completely bounded multiplier algebra $M_{cb}^{l}({{L}^{1}}(\mathbb{G}))$ to a locally compact quantum group $\mathbb{G}$, and the assignment of the intrinsic group, form functors between appropriate categories. Morphisms of locally compact quantum groups can be described by Hopf $*$-homomorphisms between universal ${{C}^{*}}$-algebras, by bicharacters, or by special sorts of coactions. We show that the whole theory of completely bounded multipliers can be lifted to the universal ${{C}^{*}}$-algebra level, and that the different pictures of both multipliers (reduced, universal, and as centralisers) and morphisms then interact in extremely natural ways. The intrinsic group of a quantum group can be realised as a class of multipliers, and so our techniques immediately apply. We also show how to think of the intrinsic group using the universal ${{C}^{*}}$-algebra picture, and then, again, how the differing views on the intrinsic group interact naturally with morphisms. We show that the intrinsic group is the “maximal classical” quantum subgroup of a locally compact quantum group, that it is even closed in the strong Vaes sense, and that the intrinsic group functor is an adjoint to the inclusion functor from locally compact groups to quantum groups.

The avoidability of binary patterns by binary cube-free words is investigated and the exact bound between unavoidable and avoidable patterns is found. All avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the growth rates of the avoiding languages are studied. All such languages, except for the overlap-free language, are proved to have exponential growth. The exact growth rates of languages avoiding minimal avoidable patterns are approximated through computer-assisted upper bounds. Finally, a new example of a pattern-avoiding language of polynomial growth is given.

The (−β)-integers are natural generalisations of the β-integers, and thus of the integers, for negative real bases. When β is the analogue of a Parry number, we describe the structure of the set of (−β)-integers by a fixed point of an anti-morphism.

We consider shifted equality sets of the form EG(a,g1,g2) = {ω | g1(ω) = ag2(ω)}, where g1 and g2 are nonerasing morphisms and a is a letter. We are interested in the family consisting of the languages h(EG(J)), where h is a coding and (EG(J)) is a shifted equality set. We prove several closure properties for this family. Moreover, we show that every recursively enumerable language L ⊆ A* is a projection of a shifted equality set, that is, L = πA(EG(a,g1,g2)) for some (nonerasing) morphisms g1 and g2 and a letter a, where πA deletes the letters not in A. Then we deduce that recursively enumerable star languages coincide with the projections of equality sets.