Galois and Post Algebras of Compositions (Superpositions)

The Galois algebra and the universal Post algebra of compositions are constructed. The universe of the Galois algebra contains relations, both discrete and continuous. The found proofs of Galois connections are shorter and simpler. It is noted that anti-isomorphism of the two algebras of functions and of relations allows to transfer the results of the modern algebra of functions to the algebra of relations, and vice versa, to transfer the results of the modern algebra of relations to the algebra of functions. A new Post algebra is constructed by using pre-iterative algebra and by adding relations as one more universe of the algebra. The universes of relations and functions are discrete or continuous. It is proved that the Post algebra of relations and the Galois algebra are equal. This allows to replace the operation of conjunction by the operation of substitution and to exclude the operation of exist quantifier.


Introduction
The Post algebra was created by E. Post in [1] (1941). He used this algebra to construct the classification of Boolean functions.
The term "Post algebra" appeared in the first in [2] (1942). But the mathematically precise definition of this algebra was given first by A. Mal'cev in [3] (1976). Mal'cev gave definitions of two algebras, which he called preiterative and iterative Post algebras. The Post algebra is the pre-iterative algebra. The iterative algebra was created by S. Jablonskij ([4], 1958). He used it to construct a partial classification of finite-valued discrete functions. Therefore, he obtained results different from Post's results at constructing the classification of Boolean functions ( [5], 1966).
The term "Galois algebra of superpositions" is introduced in this article. This term is given because this algebra was used in [6 -9] to construct the Galois connections.
Classification of all subalgebras of Post algebra was given for any k in [10] (2014).
Many statements are given without proofs or with ideas of proofs. Proofs are omitted, if they are obvious. The formalization of the remaining proofs would lead to increase the article volume and to difficulty in understanding it. Some statements and its proofs are better formalized.  The sort is: N k (the first k natural numbers), N (natural numbers), Z (integers), Q (rational numbers), R (real numbers), C (complex numbers). It is generally accepted to denote N k by k in this definition.

Galois Algebra
The members of the set S R are relations and they are denoted by r .
The fundamental operations are given over the set S R . The operations ζ and τ permute variables in relations. The operation ∆ reduces the number of variables by identifying the first two variables. The operation ∃ is an existence quantifier of the first variable. The operation & is a conjunction of two relations. A. Mal'cev ( [3]) gave the definitions of the first three operations for functions. But they are applicable to relations without changes.  1 . The real function inherits all properties of the discreet function: it belongs to maximal clone preserving 0 and to monotone clone, the clone of these functions is non-fictitious ( [11]) since it has one-membered basis } { 2 1 x x + , and so on. x to the relation 2 r . After these additions and after consequent conjunction, the relation of arity 1 2 n n + is obtained. The number of lines in this relation is 1 2 n n , and some of these lines can be equal. The first 1 n components of each line in 1 2 & r r are one of lines of 1 r , the following 2 n components are one of lines of 2 r .
Fictitious variable is an j x for which The number of all possible operations over the set S R is infinite, but all of them are constructed from the fundamental operations. Therefore, the fundamental operations are primitive (they cannot be constructed but they can construct all other operations of an algebra). It was shown above that the operation of adding a fictitious variable to a relation is not primitive.
The fundamental operations are used to create universes that are subsets of the universe S R . 1 R , -the result of applying the operations of permutation, identification and existence to members of 1 [ ] R , -the result of conjunctions of members of 1 [ ] R : This definition is iterative. At the first step of the iteration, 2 R contains all members of 1 R . At the second step, compositions of members of 2 R are added to 2 R . At the next step, the resulting set 2 R is added by compositions of its members. And so on.
A universe of functions is called a clone, if it contains the selective (projective) function 2 By the next lemmas, a set of functions preserving a relation is a clone, and a set of relations preserved by a function, is a coclone. The proof of the lemmas is given precise mathematically by using results of Mal'cev ( [3] Therefore, the result of the operations ζ and τ over functions from ( ) Pol r is a function of ( ) Pol r . If the first two variables in a function f are identifies then lines with unequal values of these variables will be removed from the table of f . The following lines will remain: But the function f continues to preserve r after removal of some lines in the matrix of r . Therefore, the result of the operation ∆ is a function of ( ) Pol r . It was necessary to prove that the result of the operation * over functions in ( ) Pol r is a function from ( ) Pol r .
Let arbitrary functions 1 f and 2 f preserve the relation r : Further it is proven that the set ( ) Inv f is a universe. A function f preserving some relation will preserve the relation after permutation of columns in the table of this relation, i.e., after permutation of lines in the matrix of the relation.
Therefore, the result of the operations ζ and τ over relations from ( ) Inv f is a relation from ( ) Inv f . If two variables in r are identified then lines with unequal values of the variables will be removed from the table of r . Therefore, corresponding columns in the matrix of r will be removed. But the remained columns have two lines to be equal.  n columns, each line of which is a line from the table 2 r . Hence, the matrix of the relation r contains columns which first 1 n lines are lines from the matrix of 1 r and the other 2 n lines are lines from the matrix of 2 r . Hence, a function f preserving 1 r and 2 r preserves 1 2 & r r too. □

Galois Connections
The set of all functions preserving the relation r is ( ) Pol r . The set of all relations preserved by a function f is ( ) Inv f , and for the set of relations R and for the set of functions F .
The Galois connection between ( ) Pol R and ( ) Inv F is given by two theorems, much simpler proofs of which are given below.
The first Galois connection is given by the following theorem. R Inv Pol R . □ The result of these two theorems is a very important Theorem 3. Diagrams of clone inclusions and of coclones inclusions are anti-isomorphic: There is a one-to-one connection between clones and coclones.
The anti-isomorphism of clones and coclones generates an anti-isomorphism of the algebra of functions and of the algebra of relations.
The main objects of mathematics are functions and relations that are objects of the algebra of functions and the algebra of relations. The main problem of these algebras is a classification of their objects. This classification is realized by diagrams of inclusions of clones and coclones. To build a classification of clones and then to classify coclones is much easier than to do the reverse -to build a classification of coclones and then to classify clones.
The anti-isomorphism of these algebras allows to transfer the results of the modern algebra of functions to the algebra of relations, and to transfer the results of the modern algebra of relations to the algebra of functions.

Post Algebra
The definition of the Post algebra for discrete functions was given by Mal'cev ( [3]). This definition is added by continuous functions and by discrete and continuous relations.

Conclusion
The Galois algebra of superpositions is constracted. The universe of the algebra contains both discrete and continuous relations. The fundamental operations of this algebra include the conjunction and the existence quantifier. Much shorter and simpler proofs of Galois connections is found (the proofs of Galois connections in the fundamental papers take several pages). It is noted that the anti-isomorphism of inclusions of clones and coclones allows a laborious classification of relations to replaced by a less laborious classification of functions. It is shown that the ultra-diagonals are redundant and can be replaced by more simpler diagonals.
The Post algebra has been constructed such that the algebra has universe containing either discrete or continuous functions or relations. Almost all researchers include the addition of fictitious variables to the Post algebra as one more fundamental operation. It is shown that this operation is not a primitive.
It is shown that the Galois algebra and the Post algebras of relations are equal. But the Post algebra is simpler and includes the algebra of functions too.