112 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Extension of 2-Dimensional Planar Systems from Homological Algebra Perspective Dr. Lewis Brewa*, Joseph Acquahb, Prof. Newton Amegbeyc a,bUniversity of Mines and Technology, Department of Mathematics, Tarkwa, Ghana cUniversity of Mines and Technology, Faculty of Engineering, Tarkwa, Ghana aEmail: amolewis@yahoo.com / lbrew@umat.edu.gh bEmail: jacquah@umat.edu.gh cEmail: na.amegbey@umat.edu.gh Abstract This paper presents 2-Dimensional (2D) planar system ( )xT together with its changes in a topological space X as a dynamical system. Continuity which is one of the topological properties can be observed from the extension of systems. This paper therefore searches for something computable or an algebraic invariant to identify this topological property of 2D planar system ( )xT . The approach is based on the notion of chains and cochains from homological algebra since the vertices and the edges are represented by the chain groups. The connectivity between different parallel chain complexes of the system is represented by the cochain groups. The extension of the 2-Dimensional planar system given by the homomorphism ( ),F b x for each parameter Zb∈ is the sequence of the cochain groups. The surface area of the system is represented by a cocycle and each cocycle is provided by a change in the parameter Zb∈ . The dynamical properties of the system are studied by analysing different cocycle over it. The novel feature is the extension of the systems using the map ( ),F b x , the cocycle and the 1st-Cohomology group which are detailed in the paper. Keywords: Cochain groups; Chain groups; Cocycle; 1st-Cohomology group; Homotopy, Dynamical System; Exact sequence; Long Exact Sequence; 2-Dimensional (2D) planar system in the space X ( )xT ; Chain complex and Cochain complex. ------------------------------------------------------------------------ * Corresponding author. http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 38, No 1, pp 112-117 113 1. Introduction The study of dynamical systems involves a wide range of methods of analysing iterated mappings. These varied methods, which are due to developments in mathematics, have made it necessary to consider the study of dynamical system from other different perspective as carried out by this paper. In this paper, the issue of the extension or changes of 2D planar system is studied via the homomorphism 1: b bF C C− → where C∗ is the cochain map of the chain groups C∗ of the system and F is the homomorphism of the cochain maps [2]. The concepts of cocycles, cochain complexes and the 1st cohomology group are used to study the connectivity of each extended system. Let X be a topological space and let ( )bC X and ( )bD X be two different horizontal parallel singular chain complexes in the space where each chain complex is connected together by a boundary b∂ . If the horizontal chain complexes ( )bC X and ( )1bD X+ are connected together by the cochain map ( ) ( )1:b b bC C X D X+→ , then the cochain complex consisting of group of homomorphisms 1: b bF C C− → for 0b ≥ where 1 1 b b b bC C− +∂ = ∂ provides a closed structure called a 2D planar system ( )xT . The closed surface area exhibited by 2D planar system is homeomorphic to 2-cell and is called cocycle. The variable 0b ≥ outlines the changes in the 2D planar system at any time, as the homomorphism ( ),F b x continues to evolve. An abstract complex or system is a formal construction that builds a space in a combinatorial way through more simple objects. In this paper, the simple objects of a system are the vertices and the edges joining the vertices. The vertices correspond to 0-chain groups and the edges correspond to 1 chain groups [1]. Therefore 2D planar system inherits a cell complex with four vertices, four edges and one 2- cell and its extension over time 0b ≥ is determined through the homomorphism ( ),F b x . 1.1 2D Planar Systems Let the cochain ( ) ( )1:b b bC C X D X+→ be a map of the chain complexes of the system in a topological space X such that the differentials b∂ and 1b+∂ are the connecting homomorphism of the chain complexes bC and 1bD + . If the two parallel chain complexes ( ) ( )1:b b bC X C X−∂ → and ( ) ( )1 1: Db b bX D X+ +∂ → are connected together by cochain bC such that 1F : b bC C− → is a map of cochain complexes and 1 1 b b b bC C− +∂ = ∂ then the combination of chain and cochain complexes as illustrated at Figure 1.0 is a 2D planar system. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 38, No 1, pp 112-117 114 Figure 1: 2D Planar System The long exact sequence of the 2D planar system increases the degree of the cochain groups. According to [3] the long exact sequence is another form of defining the extension of a short exact sequence. Thus, if 0 0f gA B C→ → → → is a short exact sequence of the cochain complexes, then there exists natural maps ( ) ( )1: d dH C H A+∂ → which provides the long exact sequence ( ) ( ) ( ) ( ) ( )1 1f Gd d d d dH C H A H B H C H A∂ ∂− +→ → → → → as the extension of the short exact sequence. Proposition 1.2 A 2D planar system is obtained if the chain complexes ( ) ( )1:b b bC X C X−∂ → and ( ) ( )1 1: Db b bX D X+ +∂ → are connected together by cochain bC such that there exists a map 1F : b bC C− → of cochain complexes and 1 1 b b b bC C− +∂ = ∂ . Proof Let X be a space containing 2D planar system and ( ) ( )1:b b bC X C X−∂ → and ( ) ( )1 1: Db b bX D X+ +∂ → be any two horizontal parallel chain complexes connected together by a cochain map bC as shown at Figure 1.0. Let bτ be a common boundary of the two triangular systems 1T and 2T .We see that each triangular system has a closed boundary. If the 2D planar system is oriented in a clockwise direction, then the boundary of 1T thus ( )1T∂ is ( ) 1 1 b b bT C τ−∂ = ∂ + − Similarly ( )2 1 b b bT Cτ +∂ = − ∂ − But the combination of the boundaries of the two triangular systems ( )1 2T T∂ + is the same as the boundary ( )xT∂ of the 2D planar system. Therefore, ( ) ( ) 1 1 2 1 b b x b b b bT T T C Cτ τ− +∂ + = ∂ = ∂ + − + −∂ − ⇒ ( ) 1 1 b b x b bT C C− +∂ = ∂ + −∂ − Thus, ( ) ( ) ( )1 1 b b x b bT C C− +∂ = ∂ + − ∂ + . But from Figure 1.0 we realised that the composites of ( )1 1b b b bC C− −∂ + = ∂ and ( )1 1 b b b bC C+ +∂ + = ∂ . However, since the closed system xT is bounded, its boundary ( )xT∂ is zero. Thus, ( ) 0xT∂ = which implies that: American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 38, No 1, pp 112-117 115 ( ) ( ) ( ) ( )1 1 1 10 b b b b b b b bC C C C− − + += ∂ + − ∂ + = ∂ − ∂ Hence 1 1 b b b bC C− +∂ = ∂ as required by the proposition. 2. The Extension of System With the 2D planar system xT , which consist of chain complexes and cochain complexes, we can form the sequence of 2D planar system. The natural homomorphism ( ) 1, : Cb bF b x C− → or ( ) 1, : Cb bF b x C +→ for 0b ≥ provides each state x of xT . For 0b = , the homomorphism ( ) 0 10, : CF x C→ provides only one and initial state of 2D planar system such that the chain and cochain complexes shown at Figure 2 commutes Figure 2: The First State of 2D Planar System Similarly, when 1b = , the extension of the system given by the homomorphism ( ) 1 21, : CF x C→ has two 2D planar systems, thus one extra 2D planar system from the initial one as shown in Figure 3. Figure 3: The Extension of 2D planar system In each state, the closed surface area is the cocycle. Therefore, the initial state has one cocycle while the second state referred to as the first extended state has two cocycles. The extension of the system is the combination of 2-cells which means that each extended part of the system consists of a number of cocycles. Thus if ( )0, 1F x = cocycle, ( )1, 2F x = , and ( )2, 3F x = etc. The initial system is based on the notion of the American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 38, No 1, pp 112-117 116 chain map 1 1 0: C C∂ → and the cochain map ( ) 0 1F 0, x : C C→ . The addition of a new cochain group is dynamically determined by the homomorphism ( ),F b x from the system’s initial state. 3. 1st – Cohomology group 1H and Extensions Considering ( ) 1, : Cb bF b x C +→ , for 0b ≥ the extension of the system in terms of the cochain complex is given as ( ) ( ) ( ) ( )3, 2, 1, 0,3 2 1 0... 0F x F x F x F xC C C C← ← ← ← ← In the form of 2D planar system the complete extended system nE for 1, 2...n = is illustrated at Figure 4. Figure 4: Extended System If nE is the extension of the system at each state for each value { }0n Z +∈ − , then from Fig. 1.3, we have ( ) ( ) 1 0, x , n n b E F F b x = =  where ( )0,F x and ( ),F b x are the inclusions in nE . Each ( ) ( ) 1 0, x , n n b E F F b x = =  is a complete bounded closed system consisting of cocycles and its respective coboundaries. Given that ( )b,KerF x is the set of cocycles or closed surface area of the cochain map bC , its coboundary formed by 1-cells is the image of ( )b,KerF x denoted by ( )Im ,of F b x where the ( ) ( )Im , b,of F b x KerF x⊂ . Since each extension nE provides the same 2D planar closed systems with different sizes, we can compute the 1st - cohomology group ( )1 nH E of system at each state. The 1st- cohomology group ( )1 nH E is the ratio of cocycle to the coboundary. Thus, the 1st–cohomology group of each extended system is given by: American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 38, No 1, pp 112-117 117 ( ) ( ) ( ) 1 b, Im ,n KerF x H E of F b x = for 1, 2,...n = and 1b ≥ . Indeed, for 2,3,...n = the system increases in size but the shape is maintained. Each state of 2D planar closed systems has a complete cocycle as explained earlier. Thus, the first enlarged 2D planar closed systems ( )xF ,1 is obtained by adding ( )xF ,0 to ( )xF ,1 to obtain one complete cocycle as shown in Fig. 1.2. Similarly, the second 2D planar closed system is obtained by the combination of ( )xF ,0 , ( )xF ,1 and ( )xF ,2 . Since each enlarged 2D planar closed systems defines cocycle and every cocycle is bounded by coboundary which is the image of ( )b,KerF x , the image of ( )b,KerF x is a divisor of the order of ( )b,KerF x . Thus the 1st –cohomology group of each 2D planar closed system, which is defined by the ratio of cocycles to the coboundary, provides a definite value, which we shall consider as Z, the group of integers. The computed value of the 1st–cohomology group of each 2D planar closed system therefore shows all the integer combination of the 2D planar closed systems. An indication that there is 1, 2…increase in continuity and connectivity of the parts of the 2D planar closed systems. This is another technique of recovering information about the continuity and connectivity of complete 2D planar closed systems through the actions of its components. The algebraic invariant, thus the 1st– cohomology group 1H provides the computable quantity for identifying the continuity and connectivity of the system. This is a new direction considered in studying the extensions of planar systems using cohomology group as the notion of continuous maps. 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