1.7. Fuentes Jurídicas de la investigación de análisis de contexto
1.7.4. El análisis de contexto en el Acto Legislativo No 01 de 2012
In this subsection we seek to establish givenm∈Z≥3andk∈Z≥m, for which combinations ofµ, γ∈Z≥0
such thatµ+γ >0 do there exist products ofm-apsis generators containing a block of type (µ, γ)?
Note a product ofm-apsis generators must contain at least one upper m-apsis and at least one lower m-apsis. Hence after excluding one upperm-apsis and one lowerm-apsis to cover cases where k <2m, which is done implicitly from this point, blocks in a product ofm-apsis generators may contain at most k−mupper vertices and at mostk−mlower vertices. Consequently we must haveµ, γ≤k−m.
For k = m the m-apsis monoid trivially consists of the identity and the single m-apsis generator am m, one way in which this may be seen is observing it is trivially the case that each m-apsis generator is idempotent, ie. (am
i )
2
=am
i for allm∈Z>0,k∈Z≥m andi∈ {1, . . . , k−m+ 1}.
Fork > mwe have seen that m-apses, which satisfy {µ, γ} ={0, m}, and transversal lines, which have µ=γ= 1, appear in products ofm-apsis generators. To form an idea of which other combinations ofµ andγ do appear we proceed by examining block types that appear in products of triapsis generators for smaller values ofkgreater than three:
(i) Whenk= 4 there are two triapsis generators, which generateA34=id4,r11,r 1 2,r 2 1,r 2 2 . Since runs
contain onlym-apses and transversal lines, no further block types appear;
(ii) When k = 5 transversals appear of type (2,2), for example in the product a3
1a33 as illustrated in
Figure 3.9. Hence for combinations of µ, γ ∈ {0,1,2} such that µ+γ >0, blocks of type (µ, γ) appear in elements ofA35 whenµ=γ; and
(iii) Whenk= 6 transversals also appear of type (3,3), for example in the producta3
1a33a32a34as illustrated
in Figure 3.10. Hence for combinations of µ, γ ∈ {0,1,2,3} such that µ+γ > 0, blocks of type (µ, γ) appear in elements ofA36 whenµ=γor {µ, γ}={0,3}.
Before examining whenk≥7, at this point it is reasonable to conjecture that, givenm∈Z≥3,k∈Z≥m andµ∈ {1, . . . , k−m}, there exist products ofm-apsis generators containing a block of type (µ, µ). 3.1.11 Definition: Letω(kµ,µ) denote the planar bipartition consisting of:
Sec 3.1:Them-apsis generated diagram monoidAmk
Figure 3.9: Given m= 3 andk= 5,a3
1a33 has a type (2,2) transversal.
a31
a3 3
=
Figure 3.10: Givenm= 3 andk= 6,a3
1a33a32a34 has a type (3,3) transversal.
a3 1 a33 a3 2 a34 =
(ii) the lower m-apsis{(µ+ 1)0, . . . ,(µ+m)0};
(iii) the (µ, µ)-transversal{m+ 1, . . . , m+µ,10, . . . , µ0}; and
(iv) the vertical lines{j, j0} wherej∈ {µ+m+ 1, . . . , k}.
For example given m= 3,ω(25,2)∈PP5 is depicted in Figure 3.11.
Figure 3.11: Givenm= 2 andk= 5,
ω(25,2)= ∈PP5
We proceed by establishing in Proposition 3.1.12 that ω(kµ,µ) ∈ Amk. The reader may find it useful to refer to Figure 3.12 which illustrates that given m = 3, k ∈ Z≥m, and λ ∈ {2, . . . , k−m}, ω
(λ,λ)
k =
ω(kλ−1,λ−1)a3λ−1a3λ+1.
3.1.12 Proposition: For each m∈Z≥3,k∈Z≥m andµ∈ {1, . . . , k−m},
ω(kµ,µ)=r12 µ Y i=2 ami−1ami+1∈A m k.
Proof. That ω(1k,1) is equal to the run r12 follows by definition. Let λ ∈ {2, . . . , µ} and suppose that ω(kλ−1,λ−1)=r12 Qλ−1 i=2 a m i−1ami+1. Note thatamλ−1a m λ+1= idλ−2⊕ω (2,2) k−λ+2 consists of:
(i) twom-apses{λ−1, . . . , λ+m−2} and{(λ+ 1)0, . . . ,(λ+m)0};
(ii) a (2,2)-transversal{λ+m−1, λ+m,(λ−1)0, λ0}; and
(iii) vertical lines{j, j0}wherej ∈ {1, . . . , λ−2, λ+m+ 1, . . . , k}.
When forming the productωλλ−−11am λ−1amλ+1:
(i) The upperm-apsis{1, . . . , m}inωλλ−−11is preserved;
(ii) The lowerm-apsis{(λ+ 1)0, . . . ,(λ+m)0} inamλ−1amλ+1 is preserved; and
(iii) The vertical line{λ+m,(λ+m)0}inωλλ−−11joins to the type (2,2) block{λ+m−1, λ+m,(λ−1)0, λ0} in amλ−1amλ+1, which joins to the lower m-apsis {λ0, . . . ,(λ+m−1)0} in ωλ−1
λ−1, which joins
to the upper m-apsis {λ−1, . . . , λ+m−2} in amλ−1amλ+1, which joins to the type (λ−1, λ − 1) block {m+ 1, . . . , m+λ−1,10, . . . ,(λ−1)0} in ωλ−1
λ−1, which joins to the vertical lines {j, j0}
in amλ−1amλ+1 where j ∈ {1, . . . , λ−2}. This forms the block {m+ 1, . . . , m+λ,10, . . . , λ0} when collecting the upper vertices in the aforementioned blocks ofωλλ−−11 along with the lower vertices in the aforementioned blocks ofam
λ−1amλ+1.
Therefore we haveω(kλ,λ) = ωk(λ−1,λ−1)am
λ−1amλ+1 = r12
Qλ
i=2ami−1ami+1, and hence it follows by induction
thatω(kµ,µ)=r12Qµ i=2a m i−1a m i+1 ∈A m k.
Figure 3.12: Givenm= 3,k∈Z≥m, andλ∈ {2, . . . , k−m},ω(kλ,λ)=ω
(λ−1,λ−1) k a 3 λ−1a3λ+1. ω(kλ,λ) = ω(kλ−1,λ−1) a3 λ−1a 3 λ+1
Sec 3.1:Them-apsis generated diagram monoidAmk
Next we seek to identify whether for any further combinations ofµ, γ∈ {0, . . . , k−m}such thatµ+γ >0, there exist products ofm-apsis generators containing a block of type (µ, γ). To do so we examine products of triapsis generators whenk≥7:
(i) Whenk= 7 the transversal types (4,1) and (1,4) appear, for example in the products ω4 4a34 and
a3 4ω44;
(ii) When k= 8 the transversal types (5,2) and (2,5) appear, for example in the products ω55a34 and a34ω55; and
(iii) When k= 9 the transversal types (6,3) and (3,6) along with the non-transversal types (6,0) and (0,6) appear, for example in the productsω6
6a34,a34ω66,ω66a34a37and a34a37ω66.
At this point it would be reasonable to conjecture that for each m ∈ Z≥3, k ∈ Z≥m, and µ, γ ∈ {0, . . . , k−m}such thatµ+γ >0 andµ≡γ (modm), there exist products ofm-apsis generators that contain a block of type (µ, γ).
3.1.13 Definition: For each m ∈Z≥3, k∈ Z≥m andµ, γ ∈ {0, . . . , k−m} such that µ+γ > 0 and µ≡γ (modm), letk= max{µ, γ}+mand letω(kµ,γ)denote the planar bipartition that contains:
(i) for eachj∈n1, . . . ,km−µo, the upperm-apsis{m(j−1) + 1, . . . , mj};
(ii) for eachj∈n1, . . . ,km−γo, the lowerm-apsis{(γ+m(j−1) + 1)0, . . . ,(γ+mj)0};
(iii) a type (µ, γ) transversalk−µ+ 1, . . . , k,10, . . . , γ0 ; and (iv) for eachj∈
k+ 1, . . . , k , the vertical line {j, j0}.
For example, givenm= 3, ω(59,2)∈PP9 is depicted in Figure 3.13.
Figure 3.13: Givenm= 3 andk= 9,
ω(59,2)= ∈PP9
We proceed by establishing in Proposition 3.1.14 thatω(kµ,γ)∈Amk. The reader may find it useful to refer to Figure 3.14 which illustrates that, given m= 3 andk= 12,ω(812,2)=ω12(8,8)θ33,,66.
3.1.14 Proposition: For eachm∈Z≥3,k∈Z≥m and µ, γ ∈ {0, . . . , k−m} such that µ+γ >0 and µ≡γ (modm),
ω(kµ,γ)= ω(kµ,µ)θγγ+1+1,γ,γ++mm+1+1,...,µ,...,µ−−mm+1+1 ifµ > γ; r1 2 Qµ i=2a m i−1ami+1 ifµ=γ; and θmm+1+1,,22mm+1+1,...,γ,...,γ−−µµ+1+1ω(kγ,γ) ifµ < γ, and henceω(kµ,γ)∈Amk .
Proof. We already established in Proposition 3.1.12 that if µ=γthenω(kµ,γ)=r12
Qµ i=2a m i−1a m i+1∈A m k. Ifµ > γthen when forming the productω(kµ,µ)θγγ+1+1,γ,γ++mm+1+1,...,µ,...,µ−−mm+1+1:
(i) the upperm-apsis{1, . . . , m}inω(kµ,µ)is preserved;
(ii) the lower m-apsis {(µ+ 1)0, . . . ,(µ+m)0} in ω(µ,µ)
k is preserved since it connects to the vertical lines{j, j0}inθγγ+1+1,γ,γ++mm+1+1,...,µ,...,µ−−mm+1+1 wherej ∈ {µ+ 1, . . . , µ+m};
(iii) the (µ, γ)-transversal{m+ 1, . . . , m+µ,10, . . . , µ0}in ω(kµ,µ) connects to
(I) them-apses{γ+ (j−1)m+ 1, . . . , γ+jm}inθγγ+1+1,γ,γ++mm+1+1,...,µ,...,µ−−mm+1+1 wherej∈
1, . . . ,µ−mγ ; and
(II) the lines{j, j0} inθγ+1,γ+m+1,...,µ−m+1
γ+1,γ+m+1,...,µ−m+1 wherej∈ {1, . . . , γ},
collectively forming the block{m+ 1, . . . , m+µ,10, . . . , γ0}; and
(iv) for eachj∈ {µ+m+ 1, . . . , k}, the vertical line{j, j0}inω(µ,µ)
k connects to the vertical line{j, j0} inθγγ+1+1,γ,γ++mm+1+1,...,µ,...,µ−−mm+1+1.
Hence ω(kµ,γ) = ω(kµ,µ)θγγ+1+1,γ,γ++mm+1+1,...,µ,...,µ−−mm+1+1 ∈ Amk. It follows analogously that if µ < γ then ω(kµ,γ) = θmm+1+1,,22mm+1+1,...,γ,...,γ−−µµ+1+1ω(kγ,γ)∈Amk.
3.1.15 Corollary: For each m ∈ Z≥3, k ∈ Z≥m and µ, γ ∈ {0, . . . , k−m} such that µ+γ > 0 and µ≡γ(mod m), blocks of type (µ, γ) do appear in some elements of them-apsis monoidAmk .
Proof. Trivially follows from Proposition 3.1.14 where we established that the well-defined bipartition ω(kµ,γ), which contains a block of type (µ, γ), may be factorised into a product ofm-apsis generators.
Sec 3.1:Them-apsis generated diagram monoidAmk
Figure 3.14: Givenm= 3 andk= 12,
ω(812,2)
=
ω(812,8)
θ33,,66
3.1.5.2 Block types that must not appear
Given m∈Z≥3, k∈Z≥m and µ, γ ∈ {0, . . . , k−m} such thatµ6≡γ (modm), it remains to establish whether there exist products ofm-apsis generators containing a block of type (µ, γ).
3.1.16 Proposition: For eachm∈Z≥2 and k∈Z≥m, them-apsis monoid Amk is a proper submonoid of the planar mod-mmonoid PMmk .
Proof. Containment follows from m-apsis generators trivially sitting inside the planar mod-m monoid
PMmk , that isam1, . . . ,amk−m+1 ∈PM
m
k . Inequality trivially follows from (2,2)-transapsis generators not being elements of the m-apsis generated monoidAmk.
3.1.17 Corollary: For eachm∈Z≥3, k ∈Z≥m andµ, γ ∈ {0, . . . , k−m} such thatµ6≡γ (modm), blocks of type (µ, γ) do not appear in any elements of them-apsis monoidAmk.