The available means of ordering and sorting in some important sections of discrete mathematics and computer science are studied, namely: in the set theory, classical mathematical logic, proof theory, graph theory, POST method, system of algorithmic algebras, algorithmic languages of object-oriented and assembly programming. The Cartesian product of sets, ordered pairs and ordered n-s, the description by means of set theory of an ordered pair, which are performed by Wiener, Hausdorff and Kuratowski, are presented. The requirements as for the relations that order sets are described. The importance of ordering in classical mathematical logic and proof theory is illustrated by the examples of calculations of the truth values of logical formulas and formal derivation of a formula on the basis of inference rules and substitution rules. Ordering in graph theory is shown by the example of a block diagram of the Euclidean algorithm, designed to find the greatest common divisor of two natural numbers. The ordering and sorting of both the instructions formed by two, three and four ordered fields and the existing ordering of instructions in the program of Post method are described. It is shown that the program is formed by the numbered instructions with unique instruction numbers and the presence of the single instruction with number 1.
The means of the system of algorithmic algebras, which are used to perform the ordering and sorting in the algorithm theory, are illustrated. The operations of the system of algorithmic algebras are presented, which include Boolean algebra operations generalized to the three-digit alphabet and operator operations of operator algebra. The properties of the composition operation are described, which is intended to describe the orderings of the operators of the operator algebra in the system of algorithmic algebras. The orderings executed by means of algorithmic programming languages are demonstrated by the hypothetical application of the modern object-oriented programming language C#. The program must contain only one method Main () from which the program execution begins. The ARM microprocessor assembly program must have only one ENTRY directive from which the program execution begins.
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