For example, the plan of electrical network for a city or the behaviour of DNA molecules in a human body can be figured out with pictures (see Figures 1, 2 and 3). Figure 1 Generating pictures of P G as given in ( 6 ). Figure 2 The single generating picture of P M given in ( 9 ). Figure 3 The single generating picture of P M in ( 12 ).
Hence, by [10], the subpicture C y, θ R with ι ( C y, θ R ) = y i μ, τ ( C y, θ R ) = y i λ and exp S ( C y, θ R ) = i μ − i λ k − l. can be depicted as in Figure 3. Figure 3 Subpicture C y, θ R of the generating picture.
To continue the study of this approach, in here, we state and prove that the presentation which has the minimal number of generators of the split extension of two finite monogenic monoids has different sets of generating functions (such that the number of these functions is equal to the number of generators) that represent the exponent sums of the generating pictures of this presentation.
Learners, generating pictures, however, seem to have less cognitive resources available for essential and generative processing, resulting in reduced comprehension.
Since we will define generating functions by considering the exponent sums of the generating pictures over the presentation of this semi-direct product, the first subsection is aimed to define these generating pictures and the related results about them.
Since we will define characteristic polynomials and generating functions by considering the exponent sums of the generating pictures over P E, the first subsection is aimed at presenting these generating pictures and the related results about them.
Proof Let us consider the generating pictures P S, x, P R, y (in Figure 4) with their non-spherical subpictures defined in Figures 2 and 3, and the generating pictures of finite monogenic monoids defined in Figure 1.
After all, the whole generating pictures P S, x and P R, y can be drawn as in Figure 4. Figure 4 Collection of the generating pictures of P M in ( 7 ).
In [13, 19], by considering the generating pictures in two different group and monoid extensions, the authors have investigated the related generating functions over the presentations of them.
If we replace z by i, then we can study the changes on the generating pictures defined in Figures 1, 2, 3 and 4. By playing on this function, one can hope to apply some operations (as defined in [7, 8]) on the pictures, and so it could happen to represent these algebraic operations by generating functions to obtain efficiency or inefficiency (while minimality holds).
The coordinates of all matching points can be extracted from the generated picture.
