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We describe a simple scheme for constructing finitely generated monoids in which left-divisibility is a linear ordering, and for practically investigating these monoids. The approach is based on subword reversing, a general method of combinatorial group theory, and connected with Garside theory, here in a non-Noetherian context. As an application we describe several families of ordered groups whose space of left-invariant orderings has an isolated point, including torus knot groups and some of their amalgamated products. [ABSTRACT FROM AUTHOR]

We construct long sequences of braids that are descending with respect to the standard order of braids (‘Dehornoy order’), and we deduce that, contrary to all usual algebraic properties of braids, certain simple combinatorial statements involving the braid order are not provable in the subsystems of the standard Peano system (although they are provable in stronger systems of arithmetic). [ABSTRACT FROM PUBLISHER]