**2. Materials and Methods**

The work is theoretical in nature, based on strict proof of the formal implications of the relevant principles and the rebuttals of the remaining formal implications, which are not satisfied. The considered formal implications relate to the defined simplified conceptual system of thermodynamics. This system consists of a set of Ω of all real or virtual processes (diagrams), along with some elementary operating rules for these processes.

An exemplary engine diagram is shown in Figure 3. For convenience, such a diagram will be marked with the arrows ↓→↓, informing respectively that: heat flows from the heater, work is performed by the device (engine), heat flows to the cooler. The arrows of such a diagram determine the signs of heat and work in the associated record of the (*Q*1, *W*, *Q*2) process, which in this case of the engine process means the positivity of all parameters: *Q*<sup>1</sup> > 0, *W* > 0, *Q*<sup>2</sup> > 0. If an arrow is pointing in the opposite direction, it means that the part of the process is going the opposite way and the corresponding heat or work symbol is negative (Figure 4). However, if a given part of the process does not occur, it is marked with 0, e.g., diagram ↓ 0 ↓ represents a process in which heat flows from the heater to the cooler without any work being done. It is assumed that the symbols of the diagrams represent (apart from the null 000 process) a continuum of processes with different values of heat and work following the arrows (and the I law of thermodynamics). In addition, diagrams specifying the efficiency of the engine (↓→↓)*<sup>η</sup>* or the efficiency of the heat pump (↑←↑)*η*˜ will also be considered.

**Figure 3.** Schematic drawing of the engine process with the equivalent arrows diagram and the equivalent notation used in the text. Most often, the engine process denoted by capital letters will denote a Carnot efficient engine. The ↓→↓ engine process with a different efficiency may then be labeled as (*Q*<sup>1</sup> ± *q*, *W*, *Q*<sup>2</sup> ± *q*) or (*q*1, *w*, *q*2).

**Figure 4.** Schematic drawing of the cooler process with the equivalent arrows diagram and the equivalent notation used in the text. The directions of the arrows show the actual direction of heat flow or how the work was done. The heat and work symbols refer nominally to the engine process, so they have negative values here.

As mentioned before, *a priori* is assumed that the diagrams satisfy the law of conservation of energy, or the I law of thermodynamics: *Q*<sup>1</sup> = *W* + *Q*2. Additionally, in the diagram set, a rule for adding diagrams is introduced, which physically means linking the processes. Adding diagrams is used, among other things, to define the concept of a convex of a set of diagrams:

#### **The definition of a convexity (for a set of diagrams—for a model)**

*A set of diagrams will be called convex if the sum of any two diagrams of this set belongs to this set. In other words, the convexity of the set requires that the add diagrams operation be internal to that set.*

Thus, the definition of the convexity of the set is here simplified to the operation of adding elements without having to consider multiplying diagrams by non-negative real numbers. However, the ability to scale diagrams means that the above definition is equivalent to the standard geometric definition of a convex set. The definition of a convex set helps to formulate the following important condition:
