La Directiva 001 de 2012 de la Fiscalía General de la Nación

2.4.28 Definition: For eachk ∈Z≥0, the Brauer monoid, which is often denoted as Bk, is the set of bipartitions such that every block contains precisely two vertices.

2.4.29 Proposition: (see [45]) For each k∈_Z≥0, the Brauer monoidBk is characterised by the gener- ators{σi,di:i= 1, . . . , k−1} along with the relations:

(i) σ2

i = idk;

(ii) σi+1σiσi+1=σiσi+1σi; (iii) σjσi=σiσj for allj−i≥2; (iv) d2i =di;

(v) didjdi =di for all|j−i|= 1;

(vi) djdi=didj for allj−i≥2;

(vii) σidi=di=diσi;

(viii) didjσi=diσj for all|j−i|= 1;

(ix) σidjdi=σjdi for all|j−i|= 1; and

(x) djσi=σidj for all|j−i| ≥2.

It turns out that Relations (vii), (viii), (ix) and (x) from Proposition 2.4.29 are stricter than required, which we establish in the following proposition.

2.4.30 Proposition: For each k ∈ Z≥0, the Brauer monoid Bk is characterised by the generators {σi,di:i= 1, . . . , k−1}along with the relations:

(i) σ2

i = idk;

(ii) σi+1σiσi+1=σiσi+1σi;

(iii) σjσi=σiσj for allj−i≥2;

(iv) d2_i =di;

(v) didjdi =di for all|j−i|= 1;

(vi) djdi=didj for allj−i≥2;

(vii) σ1d1=d1=d1σ1;

(viii) didi+1σi =diσi+1;

(ix) σidi+1di =σi+1di; and

(x) djσi=σidj either for allj−i≥2 or for alli−j ≥2.

Proof. It is sufficient for us to show that Relations (vii), (viii), (ix) and (x) from Proposition 2.4.29 may be deduced from Relations (i)-(x) above.

Leti∈ {1, . . . , k−2}. First:

Sec 2.4:Contextually relevant diagram semigroups

(ii) applying Relations (viii) and (v),di+1(diσi+1) = (di+1didi+1)σi=di+1σi; and

(iii) applying Relations (ix), (i), (viii), (v) and (i) again, σi(σi+1di)σi+1σi = (σiσi)di+1(diσi+1)σi = idk(di+1didi+1)(σiσi) =di+1.

To establish that Relation (vii) from Proposition 2.4.29 may be deduced we proceed by induction. Sup- pose thatσidi =di=diσi, noting that for induction the base case ofi= 1 is still given by Relation (vii), then we havedi+1σi+1=σiσi+1di(σi+1σiσi+1) =σiσi+1(diσi)σi+1σi=σiσi+1diσi+1σi=di+1.

It remains for us to show that Relation (x) from Proposition 2.4.29 may be deduced. First note that σi+1diσi+1=σi+1didi+1σi=σidi+1didi+1σi =σidi+1σi. Now for eachj∈ {i+ 1, . . . , k−1},

σj−1σjdj−1σjσj−1=σj−1σj−1djσj−1σj−1 =dj; and σj−1. . . σiσj. . . σi+1diσi+1. . . σjσi. . . σj−1 =σj−1. . . σiσj. . . σi+2σidi+1σiσi+2. . . σjσi. . . σj−1 =σj−1. . . σi+1σj. . . σi+2di+1σi+2. . . σjσi+1. . . σj−1 =dj.

Finally for each j∈ {i+ 2, . . . , k−1},

σidj =σiσj−1. . . σiσj. . . σi+1diσi+1. . . σjσi. . . σj−1 =σj−1. . . σiσi+1σj. . . σi+1diσi+1. . . σjσi. . . σj−1 =σj−1. . . σiσj. . . σi+1σi+2diσi+1. . . σjσi. . . σj−1 =σj−1. . . σiσj. . . σi+1diσi+2σi+1. . . σjσi. . . σj−1 =σj−1. . . σiσj. . . σi+1diσi+1. . . σjσi+1σi. . . σj−1 =σj−1. . . σiσj. . . σi+1diσi+1. . . σjσi. . . σj−1σi =djσi.

Next we establish that the relations used by Kosuda [43] on diapsis and (2,2)-transapsis generators when giving a presentation of the mod-2 monoidM2_k, outlined later in Proposition 2.4.43, also form a presentation of the Brauer monoidBk.

2.4.31 Proposition: For each k ∈ _Z≥0, the Brauer monoid Bk is characterised by the generators {σi,di:i= 1, . . . , k−1}along with the relations:

(i) σ_i2= idk;

(ii) σi+1σiσi+1=σiσi+1σi;

(iii) σjσi=σiσj for allj−i≥2;

(iv) d2_i =di;

(v) djdi=didj for allj−i≥2;

(vi) σ1d1=d1=d1σ1;

(vii) di+1=σiσi+1diσi+1σi;

(viii) d1σ2d1=d1; and

(ix) djσi=σidj either for allj−i≥2 or for alli−j ≥2.

Proof. It is sufficient for us to show that Relations (v), (viii) and (ix) in Proposition 2.4.30 may be deduced from Relations (i)-(ix) above.

First for eachi∈ {1, ..., k−1},

σi+1(di+1) = (σi+1σiσi+1)diσi+1σi =σiσi+1(σidi)σi+1σi = (σiσi+1diσi+1σi) =di+1 = (σiσi+1diσi+1σi) =σiσi+1(diσi)σi+1σi =σiσi+1di(σi+1σiσi+1) = (di+1)σi+1.

Sec 2.4:Contextually relevant diagram semigroups

Now for eachi∈ {2, ..., k−2}(note that our proof uses the first choice for Relation (ix) above),

(di)σi+1(di) =σi−1σidi−1σi(σi−1σi+1)σi−1σidi−1σiσi−1 =σi−1σidi−1σiσi+1(σi−1σi−1)σidi−1σiσi−1 =σi−1σidi−1(σiσi+1σi)di−1σiσi−1 =σi−1σi(di−1σi+1)σi(σi+1di−1)σiσi−1 =σi−1σiσi+1(di−1σidi−1)σi+1σiσi−1 =σi−1σi(σi+1di−1)σi+1σiσi−1 =σi−1σidi−1(σi+1σi+1)σiσi−1 = (σi−1σidi−1σiσi−1) =di.

Finally for each i∈ {1, ..., k−2}:

di(di+1)di= (diσi)σi+1diσi+1(σidi) = (diσi+1di)σi+1di = (diσi+1di) =di; (di+1)di(di+1) =σiσi+1diσi+1(σidi)σiσi+1diσi+1σi =σiσi+1diσi+1(diσi)σi+1diσi+1σi =σiσi+1(diσi+1di)σi+1diσi+1σi =σiσi+1(diσi+1di)σi+1σi = (σiσi+1diσi+1σi) =di+1; di(di+1)σi= (diσi)σi+1diσi+1(σiσi) = (diσi+1di)σi+1=diσi+1; and σi(di+1)di= (σiσi)σi+1diσi+1(σidi) =σi+1(diσi+1di) =σi+1di.

asdi+1σiσi+1=σiσi+1diwe get one of the relations for the singular braid monoid [5, 7]. The author notes

that in fact all of the relations for the singular braid monoid [5, 7] may be deduced from the relations in Proposition 2.4.31.