Braids and Factorizable Inverse Monoids

Abstract

What is the untangling effect on a braid if one is allowed to snip a string, or if two specified strings are allowed to pass through each other, or even allowed to merge and part as newly reconstituted strings? To calculate the effects, one works in an appropriate factorizable inverse monoid, some aspects of a general theory of which are discussed in this paper. The coset monoid of a group arises, and turns out to have a universal property within a certain class of factorizable inverse monoids. This theory is dual to the classical construction of fundamental inverse semigroups from semilattices. In our braid examples, we will focus mainly on the "merge and part" alternative, and introduce a monoid which is a natural preimage of the largest factorizable inverse submonoid of the dual symmetric inverse monoid on a finite set, and prove that it embeds in the coset monoid of the braid group.

Factorizable inverse monoids

“…The resulting presentation is usually rather prolix, but may be simpli…ed in concrete cases. For a proof, see [7]. Theorem 7.2 Suppose that U and E have monoid presentations hX U j R U i and hX E j R E i respectively; write their members as equivalence classes of words, [u] 2 U and [e] 2 E with u 2 X U , etc.…”

“…Eq (X) is complete, but K rest preserves joins (is exact) if and only if X is …nite [6], [4]. So if X is in…nite, there is no Galois adjoint.…”

“…Such a representation is not necessarily faithful, nor cofull. But each exact factorizable inverse monoid with group of units U has a minimum homomorphic image (and corresponding maximal congruence ) which embeds, as a cofull factorizable inverse submonoid, in K (U ); is also the maximum congruence which has trivial intersection with Green's relation L [6].…”

Embeddings in Coset Monoids

“…EXAMPLE 2. Let G = x be the infinite cyclic group generated by x, and let G y be the semigroup obtained by adjoining a zero y to G. Let M = (G y ) z be the semigroup [5] Embeddings in coset monoids 79 obtained by adjoing a new zero z to G y . It is easy to check that M is a factorizable inverse monoid with G M = G and E M = {1, y, z}.…”

Singular braids and partial permutations

“…Links can be simplified in a number of ways, including cutting strings, or allowing certain strings to pass through each other. Just as the closure operator on links (Alexander's Theorem [1]) led to the introduction of the braid groups [3], so too do these link simplification operations lead to new classes of algebras; examples include the inverse braid monoids IB n [12], singular braid monoids SB n [5,7], factorizable braid monoids FB n [11,15], and permeable braid monoids PB n [18]. Some of these monoids may also be thought of as "braid analogues" of various semigroups of transformations that extend the symmetric groups; for other examples of such monoids, see [17,25].…”