On monoids related to braid groups

Abstract

OUTLINE. In the first half of the thesis, we introduce the class of'chainable monoids' and describe general techniques for solving the word and division problems and for obtaining normal forms for monoids in this class. Their growth functions are shown to be rational, and calculable. We also show that a monoid with a certain central 'fundamental element' embeds in a group or related monoid. Motivating examples of monoids in this class are positive braid monoids and positive singular braid monoids. Much of the rest of the thesis is devoted to applying the techniques described to singular Artin monoids, and we obtain further results regarding conjugacy and parabolic submonoids. In the final chapter we apply the techniques of chainable monoids to braid groups of complex reflection groups. A chainable monoid is a monoid which may be defined by a finite presentation with four properties; three of which can be determined immediately from the length of the relations and their starting letters, and the fourth, the reduction property, which is more involved to verify. It has been shown that positive Artin monoids, including braid monoids, have these properties. Other examples of monoids in this class include free monoids, free commutative monoids, positive singular braid monoids (see later), a large class of one relation monoids, trace monoids and divisibility monoids. The class of chainable monoids is closed under taking direct and free products. Chainable monoids are always left cancellative. The method of chains enables the construction of a calculable partial map on pairs of words, which returns the left quotient of the first word by the second, whenever it exists, thus giving a solution to the division problem, and hence word problem, in a chainable monoid. Again using chains, we construct another partial map on pairs of words which is shown to return their least common left multiple, whenever it exists. We show that in a chainable monoid, greatest common left divisors always exist, and least common left multiples exist whenever common left multiples exist. [The author is grateful to Dietrich Kuske for recently pointing out that chainable monoids may be algebraically characterised as left cancellative monoids that