The orbit of closure-involution operations: the case of Boolean functions

For a set A of Boolean functions, a closure operator c and an involution i, let Nc,i(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{N}_{c,i}(A)$$\end{document} be the number of sets which can be obtained from A by repeated applications of c and i. The orbit O(c,i)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{O}(c,i)$$\end{document} is defined as the set of all these numbers. We determine the orbits O(S,i)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{O}(S,i)$$\end{document} where S is the closure defined by superposition and i is the complement or the duality. For the negation non\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{non}\,}}$$\end{document}, the orbit O(S,non)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{O}(S,{{\,\mathrm{non}\,}})$$\end{document} is almost determined. Especially, we show that the orbit in all these cases contains at most seven numbers. Moreover, we present some closure operators where the orbit with respect to duality and negation is arbitrarily large.


Introduction
In Kuratowski (1922), proved the following closure-complement theorem: If (X , T ) is a topological space and A ⊆ X , then at most 14 sets can be obtained from A by repeated applications of the operations topological closure and complement. Furthermore, there is a topological space and a set for which the bound 14 is achieved. More information on Kuratowski-like theorems for topological space can be found in Gardner and Jackson (2008).
Hammer (1960) noticed that such a statement holds in a more general setting; it is not necessary to consider topological spaces and topological closure. The theorem Dedicated to Prof. Gustav Burosch on the Occasion of his 80th Birthday.
Most papers related to Kuratowski's closure-complement theorem ask for upper bounds for the number of sets obtainable by repeated applications of a closure operator and complement. However, one can also consider the following more general question: Given a closure operator, determine the set of all numbers n (called the orbit of the closure operator and complement) such that there is a set A n from which we can obtain exactly n sets by repeated applications of the closure operator and complement.
In Brzozowski et al. (2009), this question was firstly investigated by Brzozowski, Grant, and Shallit for the Kleene-closure defined on formal languages and complement. They proved that the orbit of Kleene-closure and complement consists of the numbers 4, 6, 8, 10, 12, 14. Moreover, for n ∈ {4, 6, 8, 10, 12, 14}, they gave precise conditions for a language to produce exactly n languages by repeated applications of Kleeneclosure and complement.
For further language theoretic closure operators and involutions (instead of complement) the orbit was studied in Dassow (2019).
In this paper, we continue the determination of the orbit, but we consider the set of Boolean functions. Here a "classical" closure operator is defined by superpositions. The lattice of all closed sets (with respect to superpositions) of Boolean functions was determined in 1921 by Post (see Post 1921, a more complete version is Post 1941, and a modern version is Jablonski et al. 1970). We study the orbit of superpositions and complement, duality, and negation as involution. We prove that the orbit contains three, four, and at most 7 numbers for complement, duality, and negation, respectively. The corresponding Kuratowski numbers are six, four, and seven, respectively.
However, the situation changes completely if we allow other closure operators. We define some special closure operators such that with duality (or negation) the corresponding orbit contains infinity or has m elements where m is an arbitrary natural number with m ≥ 3.

Definitions and known facts
Let X be a set. We define the complement A of a set A ⊂ X by A = X \ A.
An operator c is called a closure operator on X , if the following three conditions are satisfied: An operator i is called an involution on X if, for any A ⊆ X , the relations i(A) ⊆ X and i(i(A)) = A hold.
Definition 1 Let c a closure operator on X , and i an involution on X . Then, for A ⊆ X , we define the orbit O X c,i (A) of A under c and i as the set of all sets which can be obtained from A by repeated applications of c and i and set Moreover, we define the orbit of the pair (c, i) as and the Kuratowski number of (c, i) as In this terminology, the classical Kuratowski's Theorem is given as follows: If (X , T ) is a topological space and c the corresponding topological closure, then K X (c, − ) = 14.
By the properties of an closure operator c and an involution i, in order to determine the orbit O X c,i (A) of a set A it is sufficient to determine the sets Let P 2 be the set of Boolean functions. In the rest of the paper we restrict to X = P 2 , and for the sake of simplicity we shall omit the upper index P 2 in the notations given in Definition 1.
We now define some special Boolean functions and sets of Boolean functions, which will be used later. With respect to the notation, we follow (Jablonski et al. 1970).
We use the following functions (in some cases we give two notations, and if the functions are associative, we omit some brackets in the sequel): • vel(x 1 , • sh(x 1 , x 2 ) = non(vel(x 1 , x 2 )) and sh (x 1 , x 2 ) = non(et(x 1 , x 2 )).
The dual function d( f ) of a function f is defined as Moreover, a Boolean function f is called self-dual if and only if d( f ) = f . Let D 3 denote the set of all self-dual Boolean functions. We extend the concept of negation and duality to subsets A ⊆ P 2 by setting Note that these operators non and d are involutions on P 2 .
By L 1 we denote the family of all linear functions.
By 0 < 1, an order is defined on {0, 1}. We say that an n-ary function f is monotone if and only if, for all tuples (x 1 , x 2 , . . . , x n )and (y 1 , y 2 , . . . , y n ), We now define some operations which lead to the closure operator superposition. For an n-ary function f , n ≥ 0, we set For an n-ary function f , n ≥ 2, and a permutation π on {1, 2, . . . , n}, we define (3) (π( f ))(x 1 , x 2 , . . . , x n ) = f (x π(1) , x π(2) , . . . , x π(n) ). (4) If f is an n-ary function and n ≤ 1, then we set Δ( f ) = π( f ) = f . For an n-ary function f , n ≥ 1, and an m-ary function g, m ≥ 0, we define the (n + m − 1)-ary function For a set A ⊆ P 2 , we define [A] as the set of all functions which can be obtained by finitely many iterated application of the operations defined in (1) -(5) to functions from A. It is easy to see that the operator S given by S(A) = [A] is a closure operator.

Thus, A is called closed if and only if A = [A].
We denote the set of all functions which can be obtained from f by iterated application of (1) and (2) by f .

The classical closure operator: superpositions
In this section we study the orbits of the closure operator given by superpositions and the involutions complement, duality, and negation.
Consequently, the only possible numbers which can occur in O(S, − ) are 2,4, and 6. We now prove that all these numbers are possible.
Let We now give witnesses for these numbers.
Then it is known (see Jablonski et al. 1970 If We now turn to We note that all functions in L 4 are self-dual. Let By definition, for any n-ary function from the set non( and 7 ∈ O(S, non).
It remains as an open problem whether 6 ∈ O(S, non). We conjecture that six does not belong to O(S, non). The reason for that is that there are only a few cases where six can occur (mostly we got that at most five sets are in the orbit of A), and for some of them we can show that six is impossible.

Special closure operators
In the preceding section, we have studied the orbit of superposition and some involutions. In all cases, the Kuratowski number is smaller than 7 and therefore we only get very small orbits. We shall now prove that this depends on the closure operator superposition. If we consider other closure operators on sets of Boolean functions and the involutions duality or negation, we can obtain arbitrary large Kuratowski numbers and arbitrary large orbits.

Theorem 4 There is a closure operation c
We note that the dual of a β-function is a γ -function, and vice versa. Thus d(U (β, n)) = U (γ , n) and d(U (γ , n)) = U (β, n) for n ≥ 0.
If A is not contained in V (β, n) and not contained in V (γ , n) for some n, then c 1 (A) = P 2 . Therefore c 1 (A ) ⊆ c 1 (A) is obvious.
If L is not contained in V (β, n) and not contained in V (γ , n) for some n, then c 1 (A) = P 2 . Therefore we have c 1 (c 1 (A)) = c 1 (P 2 ) = P 2 = c 1 (A).
We now determine the orbits of subsets of P 2 . Let A ⊆ V (β, n), A ∩ U (β, n) = ∅, and n even. Starting with the closure operator c 1 , we obtain the following infinite sequences of sets: c 1 (d(c 1 (L)))))) = V (γ , n + 3), . . . which proves that O c 1 ,d (A) is an infinite set. (For the sake of completeness we mention that the sequence starting with duality gives which is up to first elements the same sequence which was obtained by starting with the closure operator.) Analogously we can prove that we have infinite orbits for sets A where A ⊆ V (β, i), A ∩ U (β, n) = ∅, and n is odd or A ⊆ V (γ , i) for some n. Proof Using the notation of the preceding proof, we define c 2 by Analogously to the proof of Theorem 4, we can show that c 2 is a closure operator. If A = ∅ or A = P 2 , we obtain O c 2 ,d (A) = {A} and thus 1 ∈ O(c 2 , d).
If A is not contained in some V (β, k) and not contained in some V (γ , k), k ≥ 0, then d(A) is also not contained in some V (β, k) and not contained in some V (γ , k), and we obtain O c 2 ,d (A) = {A, d(A), P 2 } which gives 2, 3 ∈ O(c 2 , d) (as in the proof of Theorem 4). If We show this fact only for even k and even n (the proof for the other cases can be given analogously). If we start with c 2 , we get the following sets (which are obtained in succession) and if we start with d, we get d(A), V (γ , k), V (β, k), V (β, k + 1) and continue as above (where we started with c 2 ). Therefore N c 2 ,d (A) = 2(n − k + 1) + 2.
If Proof We define c 3 as follows: Now we follow the lines of the preceding proof; the only difference is that, for A ⊆ V (β, n) and A ⊆ V (γ , n) we additionally get P 2 in O c 3 ,d (A).
We mention that statements analogous to the Theorems 4, 5 and 6 also hold for the involution non. We only do the following changes: Instead of β-functions we take functions from C 2 ∩ C 3 and instead of γ -functions we take functions which are not in C 2 ∪ C 3 . Then we have the property that the negation of a function in C 2 ∩ C 3 is not in C 2 ∪ C 3 and vice versa.
We note that we cannot obtain arbitrary sets of natural numbers as orbits with respect to some closure operator and duality or negation. This comes from the fact that the following statement was shown in Dassow (2019): Let c be a closure operator on X and i an involution on X such that Consequently, because d and non satisfy the suppositions for i, certain "small" numbers have to be in O(c, d) and O(c, non).

Conclusion
In this paper we started the investigation of the Kuratowski number K(c, i)  Furthermore, we mention that it remains to study the Kuratowski number and the orbits if the basic set is the set P k of all functions which map {0, 1, 2, . . . , k − 1} n into {0, 1, 2, . . . , k − 1} for some n, i. e., of all functions of k-valued logic. By our results and their proofs (the functions q 0 (x 1 , x 2 ) = 0 f o r x 1 = x 2 x 1 + 1 mod k for x 1 = x 2 and q 1 (x 1 , x 2 ) = 1 f o r x 1 = x 2 x 1 + 1 mod k for x 1 = x 2 serve as sh and sh , respectively), we obtain O P k (S, − ) = {2, 4, 6} and K(S, − ) = 6.
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