The theory of vines

Abstract

The vine monoids are introduced as equivalence classes of configurations of strings. The strings of a vine can cross over one another, like those of a braid, and can also join together. The vine monoids map homomorphically onto the full transformation monoids. Vines in which strings do not intersect are braids. The vines in which strings only intersect form a monoid isomorphic to the monoid of order-preserving transformations of a finite chain. The vine monoid decomposes as a quotient of a general product of these two submonoids. A presentation for the vine monoids is found. The monoid of n-string vines is exhibited as a submonoid of the endomorphism monoid of the free group of rank n. Finally, some known results about the full transformation monoids are deduced.