## 1 Introduction
For example, in [12] there is a proposed IA based on new inverse operations of addition and multiplication and a new concept of the general closed interval. The large number of existing types of IA shows that they are not perfect and that some interval problems cannot yet be solved in a satisfactory and convincing manner. It also proves that the scale of difficulties associated with interval calculations, which at first glance seem very simple, is very large. A number of authors have already written about these difficulties. For example, [PERSON] in [13] draws attention to the need for a deep understanding of the interval problem to be solved and to develop equations that accurately reflect the essence of this problem. Inaccuracies in this task lead to erroneous or partially erroneous calculation results. [PERSON] in [14] draws attention to the fact that some IAs give different solutions to the problem under consideration, depending on its mathematical form that will be used. This is an unacceptable phenomenon that [PERSON] called \"unnatural behavior in modeling\" [15]. In the authors' opinion, there are still a number of not fully resolved issues in IA, which should be answered in order to improve this arithmetic. This is all the more important as IA is the basic arithmetic for the fuzzy arithmetic (FA). As [PERSON] showed in [16], fuzzy sets (FSs) can be decomposed into \(\alpha\)-cuts which are intervals. This enables operations of FA to be carried out with the aid of IA. In order to indicate the theoretical difficulties in contemporary IA, [PERSON] presented in several of his publications a certain anomaly, which will be hereinafter referred to as [PERSON]'s interval equation anomaly (LIE-anomaly or LIEA). For the first time, the LIE-anomaly was presented by [PERSON] in his keynote-lecture at Congresso Brasileiro de Sistemas Fuzzy, in Sorocaba, Brazil, in 2010. Then it was also described in other publications, e.g., recently in 2017 [17]. _[PERSON]'s Interval Equation Anomaly_  
A system is defined by the relationship \(ax=b\) in which we only have an approximate (interval) knowledge of the values \(a\) and \(b\): \(a\in[2,3]\), \(b\in[3,6]\). If we want to get knowledge about the value of the variable \(x\), we have to solve the interval Equation (2). \[[2,3]x=[3,6] \tag{2}\]This equation can be solved with 2 methods: M1 and M2. The M1 method consists in presenting the Equation (2) in the form of 2 inequalities (3) and (4). \[[2,3]x\leq[3,6] \rightarrow x\in(-\infty,1] \tag{3}\] \[[2,3]x\geq[3,6] \rightarrow x\in[3,\infty) \tag{4}\]  
Both of these inequalities yield an empty set of solution (5). \[x\in(-\infty,1]\cap[3,\infty)=\emptyset \tag{5}\]  
This solution is shown in Figure 1.  
The solution (5) obtained with the M1 method shows that the analyzed equation has no solution at all. But let's try to solve this equation by using standard interval arithmetic SIA (M2 method). Let's consider, whether there is a proper set \(X=[\underline{x},\overline{x}]\) being a solution to the equation. If it exists, then it must satisfy Equation (6). \[[2,3][\underline{x},\overline{x}]=[3,6] \tag{6}\]  
The solution of (6) obtained with the use of SIA has the form \([\underline{x},\overline{x}]=[1.5,2]\). The correctness of this solution seems to be proved by the fact that it gives equality of the left and right side of Equation (7). \[[2,3][1.5,2]=[3,6] \tag{7}\]  
Thus, solving Equation (2) with two methods M1 and M2, each of which seems to be completely logical, we obtained two different results (8) and (9), Figure 2.  
The following questions can be asked about LIE-anomalies. 1. Why are the solutions of the equation \([2,3]x=[3,6]\) obtained with the presented methods different? 2. Which of these two methods gives the correct solution? Perhaps, both solutions are correct or both are incorrect? 3. What is the meaning of the mysterious empty intervals \([1,1.5]\) and \([2,3]\) shown in Figure 2?