## 4 Multidimensional, Epistemic Fuzzy Arithmetic of Type-2 Fuzzy Numbers
The mathematical properties of type 2 fuzzy arithmetic are the same as the multidimensional T1 FA properties mentioned in Section 2 and [41]. Application of T2 FNs to Determine the Appropriate Dose of Nitrogen Fertilizer in the Cultivation of Sugar Beet  
Real world problems are often complex, in the sense that the result variable \(y\) of interest to us depends on many and sometimes very many variables \((x_{1},x_{2},\ldots,x_{n})\) which we present in the form of the functional dependence \(y=f(x_{1},x_{2},\ldots,x_{n})\). In real problems, we often have no knowledge of the values of some variables that [PERSON] [47] called information gaps (Info-Gaps). Next, we have only approximate knowledge about another part of the variables and exact knowledge we have about only some variables. The prediction of the value of the dependent variable \(y\) then takes place under the conditions of uncertainty, additionally increased by the fact that our knowledge about the mathematical form of the relationship \(y=f(x_{1},x_{2},\ldots,x_{n})\) is also usually only approximate. Therefore, we cannot demand great precision from solutions to uncertain problems. Nevertheless, we should try to solve them as best as possible, using all available knowledge. An example of a highly uncertain problem is the cultivation of crops, such as sugar beets \((SB)\). They provide a valuable break crop returning organic matter to the soil and preventing plant diseases. The beat-root cultivation starts at yield \([350,400]\)\([\text{q}/\text{ha}]\). However, using appropriate agrarian methods, some of the growers receive yield \([600,800]\)\([\text{q}/\text{ha}]\). To achieve such high yield, the skill of \(SB\) cultivation is not all that is needed. A grower also needs to be a little lucky, because the crops depend not only on the grower and his actions, but also on factors independent of humans, such as the degree of sunlight over the field in a given year (too small, small, average, high), on the temperatures (cold, warm, heat), the intensity of rainfall (no-drought, medium, too large-flooding of fields), the occurrence of plant diseases, pest infestation, etc. So, it is not only the grower's activity itself that determines the yield. However, the proper agrotechnical culture is what can influence the increase of yields. Adequate soil fertility is one of the requirements for profitable \(SB\) production. Nitrogen (\(N\)) is the most yield-influencing nutrient, and N-management is critical to obtain high \(SB\)-yield and quality. Phosphorus (\(P\)) is the next most influential nutrient, while levels of potassium (\(K\)), sulfur (\(S\)), and micronutrients usually are adequate for \(SB\) production in most soils [48]. The value of soil tests for the prediction of nutrient availability in a given area depends on how well the sample collected represents the sampled area. We should remember that the collected samples relate only to a small area and do not exactly model the whole field. The effect of nitrogen fertilization in the field has been studied for many years and is well known in terms of its quality. This influence is mainly determined by [PERSON]'s law of fertilization, [PERSON]'s maximum law and [PERSON]'s law of tolerance [49]. The typical dependence of the yield size \(P\)\([\text{q}/\text{ha}]\) on the nitrogen fertilizer dose \(N\)\([\text{kg}/\text{ha}]\) has the shape shown in Figure 31.  
Figure 30: Span function \(SP_{\mathcal{C}=\tilde{A}\tilde{B}}\) of the 5-dimensional multiplication result of T2 FNs \(\tilde{A}\tilde{B}\), of which the number \(\tilde{A}\) contains zero. The left side of the \(FOU(SP_{\mathcal{C}=\tilde{A}\tilde{B}})\) shows the inflection line corresponding to the value \(x_{\mathcal{C}}=0\). For each field, depending on the quality of its soil, sunlight, rainfall, temperatures, etc., this relationship is more or less quantitatively different, but qualitatively similar. The curve in Figure 31 has several specific ranges.