## 1 Introduction  
T2F-arithmetic relies heavily on the achievements of T1 FA. However, in the field of T1 FA, a certain chaos is currently observed, which was noticed and carefully described by [PERSON] in [1]. Currently, there are several versions of T1 FA, e.g., T1 FA based on the extension principle of [PERSON] [2], T1 FA based on alpha-cuts [3], constrained fuzzy arithmetic introduced by [PERSON] [4], LR (left border-right border) fuzzy arithmetic introduced by [PERSON] [5; 6], T1 FA based on gradual intervals [7; 8], constraint T1 FA of [PERSON] [9], multidimensional T1 FA [10; 11; 12; 13; 14; 15; 16; 17; 18], and others. [PERSON] stated in his review article [1] that in many cases different types of T1 FA provide different calculation results for one and the same problem. Sometimes, the differences in these results are significant. The question then arises, which of the T1 FAs provides the exact, or at least the more accurate result? Is it possible to check the accuracy of the result provided by some type of T1 FA, and how? In the case of T2F-arithmetic, a number of versions of this arithmetic have also been developed, such as: arithmetic of interval type 2 fuzzy numbers (IT2 FNs) [19; 20; 21; 22; 23], T2F-arithmetic using alpha-planes and quasi type-2 fuzzy numbers [24], T2 FA using the wavy slice representation [25], T2 FA based on possibility theory [26], T2 FA based on the extension principle of [PERSON] [2; 27], and others. As there are several types of T2 FAs, it is expected that, in general, they will provide different results for one and the same problem, because the calculation algorithms are different. This is just the way it is. However, this state of affairs should not be criticized too much. The emergence of the different types of T1 FAs and T2 FAs is due to the fact that the task of uncertainty calculations is far much more difficult than calculating with crisp numbers. We are currently on the hunt for the best method to do this calculation. The optimal T2 FA should be accurate,relatively easy to compute, and not too difficult to understand for engineers. Time will show which of the T2 FAs developed so far will find the greatest application in practice. T2F-arithmetic is very much needed in practice. Why? Because in many practical tasks, experts are not able to accurately determine the borders of membership functions. For example, when identifying the membership functions \"high probability\" and \"not high probability\", the expert may only give uncertain borders \(\mu_{large}^{L}(p)\) and \(\mu_{large}^{K}(p)\), as shown in Figure 1.  
Although the border of \"large probability\" is uncertain, this fact better reflects the expert's understanding of the concept. Using T2F-sets in fuzzy models and controllers increases the smoothness of their functional surfaces and enables greater accuracy or quality of control. More parameters defining T2 FS against T1 FS means more knowledge and degrees of freedom for a model or a fuzzy controller. Therefore, T2 FSs are more and more willingly used to solve practical problems. Examples of such applications: applying T2 FSs to capital budgeting [28], to multiple-criteria decision making [23; 29; 30; 31; 32; 33], in transport and to supplier selection [34; 35], to linear programming problems [20], to computing with words [22], to control problems [36], in medicine [37], in inventory management [38], or in electric problems [39]. In the future, the use of T2 FSs and T2 FA will surely increase. However, this increase will depend on the effectiveness of T2 FA. In this article, the concept of MT2 EF-arithmetic will be presented, which differs from the currently known T2 FAs in the following features. MT2 EFA provides multi-dimensional results that are the sets of all possible events (states) resulting from the uncertainty of problem inputs. This result is a universal algebraic one. It can also be called the primary (MDPR) or main result.