James F Peters

Orgest Zaka Dedicated

Girard Desargues

W Dyck

DYCK FUNDAMENTAL GROUP ON ARCWISE-CONNECTED POLYGON CYCLES IN THE DESARGUES AFFINE PLANE

Keywords: 2010 Mathematics Subject Classification. 51A30; 51E15, 51N25, 30C20, 30F40 Collinear points, Desargues Affine Plane, Fundamental Dyck Group, Simply-Connected Polygons, Skew-Field

This paper introduces the Dyck fundamental group presentation of arcwise-connected polygon cycles resulting from invariant transforms that preserve the ratio of collinear points in the Desargues affine plane. Here, a fundamental group is a collection of path-connected free groups on homotopic path cycles geometrically realized as arcwise-connected polygon cycles. For a J.H.C. Whitehead closure-finite weak (CW) topological space X, a path is a continuous map h : X × I → X over the unit interval I = [0, 1]. The geometric realization of a path is an edge in a polygon cycle, which is a sequence of edges attached to each other with no end edge. The main results in this paper, are (1) Every collection of arcwise-connected Dyck polygon cycles in the Desargues affine plane is a geometric realization of a collection path-connected homotopic path cycles. (2) Every collection of arcwise-connected clusters of triangles in the Desargues affine plane is a geometric realization of a Dyck fundamental group.

Introduction

The introduction Dyck fundamental group presentation of arcwise-connected polygon cycles in the Desargues affine plane is a outgrowth of earlier results for free groups in general [16, §3, p. 9, Theorem 4], [15, §3, p. 9] and Dyck free groups in particular [26, §5]. In this paper, edges in an affine plane are geometric realizations of homotopic paths.

Briefly, a path is a continuous map h : [0, 1] → S from the unit interval to a set of points in P in a bounded, simply-connected surface S. The surface S is simply-connected, provided every path h has end points h(0), h(1) ∈ P and h has no self-loops. A path cycle is a sequence of paths with no end path. From a geometry perspective, we have Every path has a geometric realization as an edge.

From Lemma 1, homotopic paths have a geometric realization as a polygon edge. In the context of affine geometry, the counterpart of a path cycle is a polygon cycle that is a sequence of polygon edges (called 1-cycles) with no end edge in the Desargues affine plane. For the polygons in the Desargue affine plane that we consider, every sequence of arcwise-connected vertexes in a Dyck polygon is realizable as endpoints in a homotopic path cycle in a closure-finite weak (CW) space, leading to the following result: Lemma 0.4 [15, §3, p. 9][Path Cycle Free Group Presentation]. Every path cycle in a CW space has a free group presentation.

A free group G(β, +) is nonempty finite set G of n elements equipped with binary operation (typically written as a +) and a basis β ⊂ G so that every member a ∈ G can be written as a linear combination of the generators g ∈ β, i.e., a = g∈β kg, k mod n. Free groups were introduced by W. Dyck in 1882 as a natural biproduct of cycles in simply-connected polygons P in which every vertex a ∈ P can be reached by traversing the edges between a and a distinguished vertex g ∈ P [START_REF] Dycke | Gruppentheoretischer studien[END_REF].

Results given in this paper for the Dyck fundamental group hinge on recent results concerning the association of algebraic structures in affine planes and in Desargues affine plane, and vice versa in [START_REF]Dilations of line in itself as the automorphism of the skew-field constructed over in the same line in desargues affine plane[END_REF][START_REF] Zaka | The transform of a line of Desargues affine plane in an additive group of its points[END_REF][START_REF] Filipi | The construction of a corp in the set of points in a line of desargues affine plane[END_REF][START_REF]A description of collineations-groups of an affine plane[END_REF][START_REF]Three vertex and parallelograms in the affine plane: Similarity and addition abelian groups of similarly n-vertexes in the Desargues affine plane[END_REF][START_REF] Zaka | Contribution to reports of some algebraic structures with affine plane geometry and applications[END_REF][START_REF] Zaka | Isomorphic-dilations of the skew-fields constructed over parallel lines in the Desargues affine plane[END_REF][START_REF] Zaka | Ordered line and skew-fields in the Desargues affine plane[END_REF][START_REF]Skew-field of trace-preserving endomorphisms, of translation group in affine plane[END_REF][START_REF] Zaka | The endomorphisms algebra of translations group and associative unitary ring of trace-preserving endomorphisms in affine plane[END_REF]. Those results build on the foundations for the study of the connections between axiomatic geometry and algebraic structures were set forth by D. Hilbert [START_REF] Hilbert | The foundations of geometry[END_REF]. And some classic research results in this context are given, for example, by E. Artin [START_REF] Artin | Geometric algebra[END_REF], D.R. Huges and F.C. Piper [START_REF] Hughes | Projective planes[END_REF], H. S. M Coxeter [START_REF] Coxeter | Introduction to geometry[END_REF]. Marcel Berger in [START_REF] Berger | Geometry. I, II. Transl. from the French[END_REF], Robin Hartshorne in [START_REF] Hartshorne | Foundations of projective geometry[END_REF].

preliminaries

This section briefly introduces Dyck polygon cycles and their corresponding Dyck path cycles, introduced in [START_REF] Zaka | Advances in the geometry of the ratio of linear points in the Desargues affine plane skew field[END_REF].

Definition 1. Dyck Polygon.

A Dyck polygon is a simply-connected polygon with a nonempty interior in the Desargues affine plane.

Definition 2. Dyck Polygon Cycle.

A Dyck polygon cycle is a sequence of edges with no end edge in a Dyck polygon.

Definition 3. Homotopic Path.

Let X be a simply-connected Hausdorff space in which distinct points are in disjoint neighborhoods. A homotopic path is a continuous map h : (1-cell) edge attached between polygon vertices A, P 3 (also called 0-cells on polygons P, P ′ , respectively). As a result, the boundary cyles on the polygons P, P ′ are path-connected. A, B, I on ℓ OI , A ′ , B ′ , I ′ on ℓ O ′ I ′ serve as generators on the pair of free groups on P ∪ P ′ .

[0, 1] → X defined by h(0) = p ∈ X, h(t) = p ′ ∈ X, t ∈ (0, 1), h(1) = p ′′ ∈ X [19, §2.1,p.11].
polygon P containing 3 cycles cycE 1 = A → P 3 → R → A. cycE 2 = B → P 1 → B -1 → B. cycE 3 = I → P 1 → O → I. polygon P' containing 3 cycles cycE ′ 1 = A ′ → P ′ 3 → R ′ → A ′ . cycE ′ 2 = B ′ → P ′ 1 → (B -1 ) ′ → B ′ . cycE ′ 3 = I ′ → P ′ 2 → O ′ → I ′ . The homotopic path h : [0, 1]×ℓ OI → ℓ O ′ I ′ defined by h(0, ell OI ) = A, h(t, ell OI ) = p ∈ [A, P 3 ], t ∈ [0, 1], h(1, A) = P 3 provides a

Lemma 1. Dyck Polygon Edge.

Every edge in a Dyck polygon has a homotopic path realization.

Proof. Let ℓE be an edge with endpoints p, p ′ in a Dyck polygon in a Desargues affine plane Π, which is a simply-connected Hausdorff space. Let h : [0, 1] → X ⊂ Π defined by h(0) = p ∈ int(ℓ), h(t) ∈ int(ℓ), t ∈ (0, 1), h(1) = p ′ ∈ int(ℓ), which is a homotopic path realization of ℓE.

Definition 4. Path Cycle.

A path cycle is a sequence of homotopic paths with no end path.

Definition 5. Dyck Path Cycle.

A Dyck path cycle is a path cycle with nonempty interior is a realization of a Dyck polygon cycle.

Theorem 1. [Dyck Polygon Cycle Realization]. Every Dyck polygon cycle in the Desargues affine plane has a path cycle realization.

Proof. Let cycE be a Dyck polygon cycle. From Lemma 1, every edge in cycE has a corresponding Dyck path. Hence, from Def. 5, cycE has a Dyck path cycle.

Conversely, path cycles on the boundary of a path triangle are geometrically realized as 1-cycles, i.e., Definition 6. A path triangle is a collection of path-connected vertices on three edges with no end-edge.

Every path triangle is geometrically realized as a 1-cycle (See Theorem 2). [START_REF] Whitehead | Combinatorial homotopy. I[END_REF]. A 1-cycle E is a collection of path-connected vertexes (0-cells) on edges (1-cells)

Definition 7. 1-Cycle

(s) = ∞ -∞ x(t)e -st
dt, which is the Laplace transform defined in terms of input signal x(t) and s = σ + jω, which is the complex frequency of e -st .

Transforms which Preserve Ratio of Collinear Points

In this section we prove that the parallel projection, translation and dilatation of a line in itself or in isomorphic line in a Desargues affine plane, preserving: the ratio of of two and three collinear points. The geometrical interpretations, even in the Euclidean view, are quite beautiful in the above theorems, regardless of the rather rendered figures. This is also the reason that we are giving the proofs in the algebraic sense. So we will always have in mind the close connection of skew field and a line in a Desargues affine plane, and the properties of parallel projection, translations and dilatations. For a characterization of dilatations in general, see [3, vol I, §2.5.6, p. 51

]. Briefly, a dilatation is a bijection f : X → X of an affine space X of dimension ≥ 2. Let D be any line in X, then f (D) is a line in X parallel to D.
Theorem 2. [START_REF]Progress in invariant and preserving transforms for the ratio of collinear points in the desargues affine plane skew field[END_REF] The parallel projection between the two lines ℓ 1 and ℓ 2 in Desargues affine plane, preserving the ratio of 2-points, P P (r(A : B)) = r(P P (A) : P P (B))

Theorem 2 leads to the following result based on what we know about Dyck polygon cycles.

Proposition 1. The cycles on the boundaries of the triangles in a parallel projection between a pair of lines that preserve the ratio of collinear points in the Desargues affine plane have a path cycle realization.

Definition 8. (Alexandrov Nerve [15]).

A nerve complex NrvE in a CW space is a collection of nonempty cell complexes with nonvoid intersection. This is an example of an Alexandrov nerve [1, §4.3,p. 39].

Example 3. From Prop. 1, the pair of Dyck polygon cycles in Fig. 3 have a path cycle realization. If we are working with a path cycle realization of the Dyck polygon cycle, we can always revert back to the corresponding cycle in the Dyck polygon (from Lemma 0.1). From Def. 5, Dyck polygon cycles have nonempty interiors.

With either the addition or the multiplication of points in Fig. 3, the resulting polygon cycles overlap. From Def. 8, overlapping Dyck polygon triangles in a Desargues affine plane form an Alexandrov nerve [START_REF] Peters | Homotopic nerve complexes with free group presentations[END_REF]. This is important, since Dyck polygon cycles on the boundary of a nerve have a concise presentation as free groups. Theorem 3. [START_REF]Progress in invariant and preserving transforms for the ratio of collinear points in the desargues affine plane skew field[END_REF].

Translations in the Desargues affine plane, preserve the ratio of a pair of collinear points A, B, resulting in, φ(r(A : B)) = r(φ(A) : φ(B))

Theorem 3 leads to the following result.

Proposition 2. Translations of pairs of collinear points in the Desargues affine plane result in dual Dyck path cycles.

Proof. From Theorem 3, a number of dual polygons result from the translation of pairs of collinear points in the Desargues affine plane. From Theorem 1, each sequence of edges on the dual polygons have a realization as dual path cycles, which is the desired result.

Example 4. From Prop. 2, the trio of Dyck polygon cycles in Fig. 2 have a path cycle realization, e.g.,

CycE 1 ′ is the dual of CycE 1 . CycE 2 ′ is the dual of CycE 2 . CycE 3 ′ is the dual of CycE 3 .
With the translation of collinear points in in Fig. 2, the resulting dual polygon cycles overlap. To see this, consider

CycE 3 ∪ CycE 3 ′ ∪ ℓ OP1 ,

Fundamental Dyck Group on Arcwise-Connected Polygon Cycles in the Desargues Affine Plane

This section introduces the Dyck fundamental group, which is a collection of path-connected free path groups. Each free path group is a presesntation of a polygon cycle. And each fundamental group is a presentation of a collection of arcwise-connected polygon cycles resulting from transforms that preserve the ratio of collinear points in the Desargues affine plane.

Each polygon cycle is a geometric realization of a free path group. Here, the term path means homotopic path defined by a continuous mapping h : X × I → X for points x in a space X over the unit interval I = [0, 1]. In this work, a fundamental group G is introduced by inserting one or more paths attached between free path groups {G}. The geometric realization of {G} is a Dyck polygon cycle found in collections of arcwise-connected polgon cycles that are themselves Each Dyck polygon cycle is a geometric realization of a path cycle. Planar path cycles live in a closure-finite weak topological space introduced by J.H.C. Whitehead [START_REF] Whitehead | Combinatorial homotopy. I[END_REF] (see Def. 14). The fundamental group introduced here is named after W. Dyck, who, in 1882, introduced free groups derived from polygon cycles [START_REF] Dycke | Gruppentheoretischer studien[END_REF]. Polygons are simply connected, provided the polygon vertices are in a sequence of edges with no self-intersections.

Remark 2. Poincaré Group. The Poincaré Group (aka Fundamental Group) G was named after Henri Poincaré by Herman Minkowski in 1908. For Minkowski, a Poincaré group G (actually, several path-connected groups) is the totality of all Lorentz transformations [14, p. 484].

Free groups were introduced by W. Dyck in 1882 as a natural biproduct of cycles in simply connected polygons P in which every vertex a ∈ P can be reached by traversing the edges between a and a distinguished vertex g ∈ P [START_REF] Dycke | Gruppentheoretischer studien[END_REF].

Definition 11. Dyck Polygon.

A Dyck polygon is a simply-connected polygon in the Desargues affine plane.

Definition 12. Finite Free Group.

A free group G(β, +) is a nonempty finite set G of n elements equipped with a binary operation (typically written as a +) and a basis β ⊂ G so that every member a ∈ G can be written as a linear combination the generators g ∈ β, i.e., a = g∈β kg, k mod n.

There is a path cycle associated with every Dyck polygon cycle.

Definition 13. (Path).

Let X be a set of points in a Hausdorff space, i.e., distinct points live in nonintersecting neighborhoods. A path is a continuous map h :

X × I → X with h(x, 0) = h(x 0 ) = initial point in path h. h(x, t) = h(x t ) = interior point in path h. h(x, 1) = h(x 1 ) = final point in path h.
The geometric realization of a path h is denoted by |h|. Here, |h| is an edge in a Dyck polygon.

Definition 14. (Path Cycle).

Let X be a set of points in a Hausdorff space. A finite path cycle h (denoted by hcycE) is a collection of n -1[n] ([n] denotes mod n) paths such that the initial cell h 0 (x, 0) = h n-1[n] (x, 1), i.e., path h begins and ends with the same 0-cell. Put another way, a path cycle is a sequence of paths with no end path (the first path in the cycle is also the last path in the cycle). The geometric realization of hcycE (denoted by |hcycE|) is a Dyck polygon cycle in a Desargues affine space.

Example 6. For a sample finite polygon cycle cycE which is the geometric realization |hcycE| of a path cycle hcycE, see Fig. 3. Here, vertex v 0 is the initial vertex in |hcycE| is defined by h 0 (x 0 ) = h (n-1)[n] (x 1 ), the end point in path h (n-1)[n] is the initial point in path h 0 . This path cycle has [n] = n cells geometrically realized as [n] = n vertexes in the polygon cycle cycE in the Desargues affine plane.

The focus here is on collections of path-connected free path groups G = {G(β, +)} such that each G ⊂ G is a collection of paths {h} in a path cycle hcycE. A path is a continuous map h : X × I → X,β ⊂ G. A path cycle hcycEis sequence of paths h 0 , . . . , h (n-1) [n] in which the last path h (n) [n] is also the initial path h (0)[n] , i.e., hcycE has no end path. The geometric realization of a path h is denoted by |h|. A path h defined by

X = cell complex in a CW space. x ∈ X. h(x 0 ), h(x 1 ) = 0-cells (vertices) in a CW space. > h(x 0 ), h(x 1 ) = arc in a CW space. t ∈ (0, 1) open temporal interval. h(x t ) = point at time t in > h(x 0 ), h(x 1 ), h(x 0 ) = initial 0-cell in a path, h(x 1 ) = end 0-cell in a path, |h| = geometric realization of cycle path h, h i ∈ hcycE = i th path in cycle hcycE, [n] = mod n ∈ Z + , h 0 , . . . , h (n-1)[n] = 0, ..., (n -1)[n] paths in path cycle hcycE, h i+k[n] (x t ) → h i+k[n] (x t ) move forward k vertices also, written + kh i (x t ) = kh i+k[n] (x t ) h i-k[n] (x t ) = h i-k[n] (x t ) move backward k vertices also, written -kh i (x t ) = |-kh i (x t )| h i+0 (x t ) = |h i+0 (x t )| no move also, written 0h i (x t ) = |0h i (x t )| .
Lemma 3. [START_REF] Zaka | Advances in the geometry of the ratio of linear points in the Desargues affine plane skew field[END_REF].

Every vertex in a Dyck polygon containing collinear ratio points can be reached by a finite sequence of moves from a ratio point.

Theorem 7. [START_REF] Zaka | Advances in the geometry of the ratio of linear points in the Desargues affine plane skew field[END_REF].

Every vertex in a Dyck polygon containing colinear ratio points is a linear combination of the ratio points.

Definition 15. Dyck free group move operation +. A Dyck free group denoted by G(β, +) is a collection of continuous maps h : G×I → |G| ∈ cycE, I = [0, 1] and a binary operation + + : h(G, I) × h(G, I) → |G|

defined in terms of maps that are geometrically realized as moves from one vertex to another one in a polygon cycle cycE = |G| (geometric realization of G) and basis β ⊂ G. From Theorem 7, every vertex in G is a linear combination of the elements of the basis β. For a Dyck free group G(β, +), we have Theorem 7). Here, every edge is a Dyck polygon is a geometric realization of a homotopic path. Given generators g, g ∈ β ⊂ G (set of paths), we reach vertex p in Fig. 7.1 by moving forward over the sequence of paths that ends in vertex (0-cell) p. For example, let h i := g, h j := g ′ and move ahead k edges after edge h i and then move ahead k ′ edges beyond edge h j , then we have

G = {h} . |G| = cycE. X = set of 0-cells in a CW space. h : X × I → |h| defined by h(x, 0) → |h(x, 0)| = v 0 ∈ cycE, h(x, 1) → |h(x, 1)| = v 1 ∈ cycE, h(x, t) → |h(x, t)| ∈ > |h(x, 0)| , |h(x, 1)|. 0 = |h i (∅, t)| = |∅| ∈ cycE. h, h ′ ∈ G. basis β ⊆ G. generator g ∈ β. + : G × G → G, defined by +(h, h ′ ) → ki∈Z gj ∈β k i g j + k i ′ ∈Z g j ′ ∈β k i ′ g j ′ (from
→ |k 1 h 1 + • • • + k n h n + k 1 ′ h ′ 1 + • • • + k ′ m h ′ m | geom. realiz. of lin. comb. of g ∈ β → moves over |hcycE| → h 1+k1 h 1 + • • • + h ′ 1+k ′ 1 h 1 + • • • + h ′ m+k ′ m . +(h i , ∅) = zero moves from h i ∈ hcycE → |h(x i+0 )| = |h i | .
p = kg + k ′ g = kh i + k ′ h j = h i+k + h j+k ′ .
2 o Inverse paths are illustrated in Fig. 7.2.

3 o The identity element ∅ is illustrated in Fig. 7.3.

Lemma 4. [START_REF] Zaka | Advances in the geometry of the ratio of linear points in the Desargues affine plane skew field[END_REF].

Let G(β, +) be a Dyck polygon G with basis β and equipped with move +. The structure G(β, +) is a free groupoid.

Lemma 5. [START_REF] Zaka | Advances in the geometry of the ratio of linear points in the Desargues affine plane skew field[END_REF].

Let G(β, +) be a free groupoid. Then G(β, +) is a free semigroup.

Definition 16. Herstein Identity element.

A Dyck free group (aka path group) G on a path cycle hcycE is a set of paths. The ∅ ∈ is a member of a path group G and serves an identity element. In a free group G presentation of a polygon path cycle cycE, the identity element ∅ is a very special element in G representing a path with zero length. The identity element ∅ satisfies

+(h i , ∅) = h i+0 = h i → |h i | ∈ |G| .
Lemma 6. [START_REF] Zaka | Advances in the geometry of the ratio of linear points in the Desargues affine plane skew field[END_REF].

Every Dyck free semigroup has an identity element.

Lemma 7. [START_REF] Zaka | Advances in the geometry of the ratio of linear points in the Desargues affine plane skew field[END_REF].

Every member of a Dyck free semigroup has an inverse. Theorem 8. [START_REF] Zaka | Advances in the geometry of the ratio of linear points in the Desargues affine plane skew field[END_REF]. G(β, +) is a free Dyck group.

Theorem 9. [START_REF] Zaka | Advances in the geometry of the ratio of linear points in the Desargues affine plane skew field[END_REF].

Every Dyck polygon resulting from the transforms that preserve the ratio of co-linear points in simply-connected polygons in the Desargues affine plane has a free group presentation.

Proof. Immediate from Theorem 8.

From Theorem 9, we arrive at a basis for the Dyck fundamental group on collections of path-connected Dyck free groups derived from arcwise-connected cycles on transforms that preserve the ratio of co-linear points in simply-connected polygons in the Desargues affine plane.

Definition 17. Dyck Fundamental Group.

A Dyck fundamental group (denoted by G) is a space of paths containing pathconnected Dyck free groups in the Desargues affine plane. One free group G(β, +) ∈ G is a path cycle. For each pair of free groups G, G ′ in G, there is a path h ∈ G attach between G, G ′ , forming another path cycle.

Here, |h| denotes the geometric realization of a path h. Lemma 8. Every edge in a Dyck polygon is a geometric realization of a path.

Proof. Let X be a set of points in a Dyck polygon, x ∈ X and let h : X × I → X be a path. For a path h, we obtain h(x, 0), h(x, 1), the endpoints of edge x and h(x, t) are the points in the interior of x. Hence, the geometric realization of h is |h|. Remark 3. From Def. 14, a finite path cycle is a sequence of paths with no end path. Hence, a free group G defined on a finite path cycle hcycE with n is finite, defined by G = {h i : h i ∈ hcycE}. The order of G is n. The basis β of G is a set of one or more generators h i . For example, consider β = h 0 . The rank of G (number of generators) then equals 1. Then every point h i = kh 0 with the usual + (move) operation for G. Then we have G(β, +) = G({h 0 } , +)

The geometric realization of a path cycle hcycE is denoted by |hcycE|.

Lemma 9. Every Dyck polygon cycle is a geometric realization of a path cycle.

Proof. Let hcycE be a Dyck path cycle. From Lemma 8, every edge in cycE is a geometric realization of a path in hcycE. Hence, cycE = |hcycE|. This gives the desired result.

Theorem 10. Every collection of arcwise-connected Dyck polygon cycles is a geometric realization of a collection of path-connected homotopic path cycles.

Proof. Immediate from Lemma 9.

Remark 4. Each Dyck free group is a presentation of a path cycle group geometrically realized as a polygon cycle in the Desargues affine plane. A pair of Dyck free groups G,G' is path-connected, provided there is a path h with initial vertex h(x, 0) ∈ G and with ending vertex h(x, 1) ∈ G ′ . And every collection of arcwiseconnected Dyck polygon cycles is the geometric realization of a fundamental group. Example 8. Let cycE, cycE ′ be the pair of boundary cycles on the simply-connected polygons P, P ′ in in Fig. 2

. And let

A ∈ cycE = |hcycE| , A ′ ∈ cycE ′ = |hcycE ′ | and let A ∈ ℓ 0I = |h(x, 0)| (geometric realization of h(x, 0)). A ′ ∈ ℓ 0 ′ I ′ = |h(x, 1)| (geometric realization of h(x, 1)). ℓ Op1 = |h| (geometric realization of path h).
Let G be a free group presentation of cycE and let G ′ be a free group presentation of cycE ′ . We then have G = G ∪ G ′ ∪ h. Hence, from Def. 17, G is a fundamental Dyck group. From a Dyck fundamental group presentation of a collection of path-conected polygon cycles on transforms that preserve the ratio of collinear points in the Desargues affine plane, we obtain a concise means of identifying the properties of collections of path-connected Dyck free groups, e.g., Property-1 o Fundamental Dyck Group Depth.

Every collection of n arcwise-connected polygon cycles on transforms that preserve the ratio of collinear points in the Desargues affine plane has a Dyck fundamental free group presentation, containing n free groups, which is the depth of the fundemantal group.

Property-2 o Views of arcwise-connected clusters of triangles.

Every collection T of arcwise-connected cluster of triangles is a Dyck polygon. From Lemma 11, T has a view in the form of a fundamental group.

Property-3 o Cycle Traversal.

In a Dyck fundamental group G that presents a collection of pathconnected Dyck polygon cycles, traversal of a path cycle maps to a corresponding geometric realization of a traversal over a polygon cycle. Property-4 o Path Measure.

Every vertex in a collection of path-connected Dyck polygon cycles with collinear ratio vertices is a geometric realization of a linear combination of the generator paths in the basis β in a fundamental group G(β, +). For paths h, h ′ ∈ G written as linear combinations of generators g j , g j ′ ∈ β, a path measure is the sum

k 1 + • • • + k n + k 1 ′ + • • • + k ′ m in +(h, h ′ ) → ki∈Z gj ∈β k i g j + k i ′ ∈Z g j ′ ∈β
k i ′ g j ′ (from Theorem 7).

→ |k 1 h 1 + • • • + k n h n + k 1 ′ h ′ 1 + • • • + k ′ m h ′ m | . Property-5 o Concise Presentation.
From Theorem 11, every collection if path-connected Dyck polygon cycles resulting from the transforms that preserve the ratio of collinear points in simply-connected polygons in the Desargues affine plane has a Dyck fundamental group presentation, which gives a concise view of the corresponding path cycles. Property-6 o Dyck-Feynman Path.

From an application perspective, Theorem 11 is significant, since we can then view a discrete Feynman path [6, p. xiv] (a trace of a trajectory of a particle) having a geometric realization as a sequence of edges (1-cells) in a collection of path-connected Dyck polygon cycles resulting from the transforms that preserve the ratio of collinear points in simply-connected polygons in the Desargues affine plane. Property-7 o Feynman Path Theorem. The value of a path between points a and b in a Dyck fundamental group G (i.e., the positions of a particle trajectory at times t a , t b , respectively), is K(b, a), defined in a complex space CS with respect to Planck's constant by Theorem 12. (Feynman Path Integral Theorem [START_REF] Peters | Discrete and continuous curved surface antipodal paths[END_REF]).

There exists a Feynman path with an initial path integral K(b k , x a ) for an initial vector ⃗ x a that equals the path integral K(b k , -x a ) for a later vector -⃗ x a , which may or may not be the antipode of vector ⃗ x a .

Property-8 o From Theorem 12, every Dyck fundamental group presentation has a corresponding Feynman path integral.

1. 1 :Figure 1 .

 11 Figure 1. Polygon cycles resulting from the addition and multiplication of points.

Example 1 .

 1 Sample polygon cycles that are natural outcome of the addition and multiplication of points in the Desargues affine plane are shown in Fig. 1. In this example, each polygon cycle is a sequence of edges on a triangle such as 1 o In Fig. 1.1,we have polygon cycle CycE 2 on triangle △A, P 1 , C = A + B as well as polygon cycle CycE 1 on triangle △O, B 1 , B. 2 o In Fig. 1.2,we have polygon cycle CycE 2 on triangle △A, P 1 , C = A • B as well as polygon cycle CycE 1 on triangle △I, B 1 , B.

Figure 2 .

 2 Figure 2. Translations in the Desargues affine plane

Figure 3 .

 3 Figure 3. Ratio of 2-Points in a parallel projection in the Desargues affine plane with resulting Dyck polygon cycles

  which form a collection of path-connected cycles. Further, cycles CycE 3 , CycE 3 ′ have concise presentations as free groups, which are path-connected. This pathconnectedness results line ℓ OP1 attached between the pair of cycles. Taken together, these path-connected cycles provide a simple example of a Dyck fundamental group (for more about this, see Section 4). The polygon cycles in Fig.2have nonempty interior represented by the shaded areas. From Def. 8, a pair of cell complexes with nonempty intersection form an Alexandrov nerbe. Hence, the dual overlapping cell complexes in Fig.2are examples of dual Alexandrov nerves.

Figure 4 .

 4 Figure 4. Dilatation of collinear points in Desargues affine plane.

Figure 5 .Lemma 2 .Theorem 5 .Definition 9 .

 5259 Figure 5. Ratio of 2-Points into Dilation with fixed point in ℓ OI -line, in Desargues affine plane; (a) case where V = O and (b) case where V ̸ = O.

3 o 2 - 5 oTheorem 6 .Example 5 .

 2565 cells(Lefschetz [13]). A 2-cell is a pair of arcs (1cells) with shared endpoints. 4 o triangle 2-cells (Whitehead [20]). A triangle 2-cell is a Lefschetz 2-cell with an added vertex on one of the 2-cell arcs. This is shown in . The addition of a third 0-cell in a Leftschetz 2-cell constructed with a pair of arcs a 1 , a 2 results in arc a 3 to form a round triangle with sides that are arcs. The geometric realization of a complex K is denoted by |K|. Definition 10. Cycle Betti Number. The size of the basis for a free group presentation of Dyck cycle is its Betti number. Lefschetz Polyhedra Theorem[12, S II.5]. The Betti numbers of [cell complex] K are topologically invariant in the sense that if two polyhedra |K| , |L| are homeomorphic, the the complexes K, L have the same Betti number. Geometric Structures from planar Leftschetz cell complexes. Dual Dyck polygon cycles like those in the dilatations in Fig. 4 and Fig. 5 are examples of geometric structures that are collections of Letschetz cell complexes attached to each other. From Theorem 6, the dual Dyck polygon cycles are examples of polyhedra that have the same Betti number. This is important, since this gives us a concise means of characterizing dual Dyck polygon cycles.

Figure 6 .

 6 Figure 6. |hcycE| = geometric realization of path cycle hCycE as a Dyck polygon cycle, with h 0 (x 0 ) = h (n-1)[n] (x 1 ), [n] = mod n.

Figure 7 .Example 7 .

 77 Figure 7. Geometric realization of summing paths.

Lemma 10 .Theorem 11 .

 1011 Dyck Fundamental Group on Path-Connected Dyck Free Groups. Every collection of path-connected Dyck free groups has a Dyck fundamental group presentation. Proof. Let G be a collection of Dyck free groups on polygon cycles containing a path between each pair of free groups in G. Hence, from Def 17, the presentation of G is a fundamental group Lemma 11. Dyck Fundamental Group on Path-Connected Triangle Clusters in the Affine Desargues Plane. Every collection of arcwise-connected clusters of triangles in the affine Desargues plane has a Dyck fundamental group presentation. Proof. Each cluster of arcwise-connected triangles {△} in the Desargues affine plane is a Dyck polygon. Hence, from Theorem 9, {△} has a free group presentation G. Each arcwise-connected collection of clusters of arcwise-connected triangles T is also a Dyck polygon. Again, from Theorem 9, T has a free group presentation G. Hence, from Lemma 10, T has a Dyck fundamental group presentation. Every collection of arcwise-connected clusters of triangles in the affine Desargues plane is the geometric realization of a Dyck fundamental group. Proof. Immediate from Lemma 11.

  Feynman and Hibbs [7, p. 45] by V (x, t) = Potential energy of particle x ∈ G with mass m. L = m 2 ẋ2 -V (x, t)(Lagragian for the system). S = b ta L( ẋ, x, t)dt. a, b = points in path-connected cycles presented by G. K(b, a) = b a e ( i )S[b,a] Dx(t).

Remark 1. (Application: Path Cycles as Electrical Networks).

  attached to each other with no end vertex.

	,
	Theorem 2 [17, §2, p. 40](Path Triangle Geometric Realization).
	Every path triangle has a geometric realization as a 1-cycle.
	From Theorem 2, every Dyck path cycle on a sequence of 3 paths with no end
	path has a geometric realization as a 1-cycle, which represents a simple electrical
	circuit containing 3 nodes. In general, collections of overlapping Dyck path cycles
	lead to an application in Lefschetz electrical networks (see Remark 1) concisely
	represented as collections of nodes on path-connected Dyck polygon cycles with
	derivable signals at each of the nodes.
	From S. Lefschetz [13, p. 4], an application of Dyck path cycles in terms of electrical
	networks is possible. That is, from Theorem 2, every Dyck path cycle on a sequence
	of 3 paths with no end path has a geometric realization as a 1-cycle. In general, col-
	lections of overlapping Dyck path cycles lead to an application in electrical networks
	concisely represented as nodes on path-connected Dyck polygon cycles with derivable
	signals at each of the nodes, provided cycle nodes are connected to a distinuished
	node have a known input signal x(t) at time t and each of the other cycle nodes have
	signals derivable from the flow of current through the network, having, for example,
	an output X

Acknowledgements.

The research has been supported by the Natural Sciences & Engineering Research Council of Canada (NSERC) discovery grant 185986, Instituto Nazionale di Alta Matematica (INdAM) Francesco Severi, Gruppo Nazionale per le Strutture Algebriche, Geometriche e Loro Applicazioni grant 9 920160 000362, n.prot U 2016/000036 and Scientific and Technological Research Council of Turkey (TÜBİTAK) Scientific Human Resources Development (BIDEB) under grant no: 2221-1059B211301223.