On Addition of Sets in Boolean Space

In many problems of combinatory analysis, operations of addition of sets are used (sum, direct sum, direct product etc.). In the present paper, as well as in the preceding one [1], some properties of addition operation of sets (namely, Minkowski addition) in Boolean space n B are presented. Also, sums and multisums of various “classical figures” as: sphere, layer, interval etc. are considered. The obtained results make possible to describe multisums by such characteristics of summands as: the sphere radius, weight of layer, dimension of interval etc. using the methods presented in [2], as well as possible solutions of the equation X Y A + = , where n X Y A B ⊆ , , , are considered. In spite of simplicity of the statement of the problem, complexity of its solutions is obvious at once, when the connection of solutions with constructions of equidistant codes or existence the Hadamard matrices is apparent. The present paper submits certain results (statements) which are to be the ground for next investigations dealing with Minkowski summation operations of sets in Boolean space.


Sum of Sets According to Minkowski
where ⊕ is the mod 2 addition operation.This addition operation for members of n B can be extended in subsets of n B .
In other words, if , Thus, the sum of subsets X Y + is consisted of sums of points belonging to X and Y, respectively.Examples.
is the "shift" of the set X to the point y, and X Y X + = .2. if X is a subset in n B , then X X X + =.Also, { } X Y + can be interprated as union of "shifts" of the sets X onto points of the sets Y.
The family ( ) + , with an introduced Minkowski addition operation "+" forms a monoid with the neutral element { } 0 n , which is one member set having the zero element of n B .
The following inequality is valid: Both limits are achievable here.The following statements describe the sets in which these limits are achieved.
Definition [2].The pair ( ) with shifts transferring the subset into its "shift".Definition [3].A stabilizer of the set X with respect to the group G is the union of "shifts" X G from G, con- serving X, i.
is a group of shifts, then for the following is valid: In this case X has a non-obvious stabilizer if all constituents of X can be partitioned into the pairs ( ) Thus, all subsets of X, having a non-obvious stabilizers, are described above.
In the general form the stabilizer X G for an arbitrary group G and an arbitrary set n X B ⊆ can be described in the following terms [3].
Statement 3. The constituent X g G ∈ if the set X can be partitioned into the pairs ( ) v v in such a way that i j v v g + = for all pairs which are included in the partition.This statement can be obtained by analogical consideration for n G B = as in [4].From the above statement one can construct the following algorithm for building the stabilizer X G of an ar- bitrary set X for the subgroup 3. Then we build all partitions in A out of these pairs.

P
is the set of all partitions of X having the same weights in pairs x C ∈ , then This means that all pairs ( ) ( ) ( ) ( ) ( ) ( ) , , , , , , , , , , , have the multiplicity 2 in the sum X X + .Then we have: The sum of the pairs in each of the solutions is the same.Hence, the following set: Below we present the simple properties of the operation "+"-it is addition in the sense of Minkowski, as was mentioned above-which can be taken as properties of an algebraic system with basic set 2 n B and those for operations of addition, union of sets, set intersection etc.
1. Assosiativity: 3. Distributivity with respect to union: ( ) There are finitely many other relations connecting constituents of the algebraic system described above.

Sum of Spheres in B n
Let ( ) ( ) ( ) ( ) ( ) ( ) is the set complement of the sphere B and v is the logic "negation" of the binary set v. We assume that ( ) ( ) p S M be the set of points belonging to the union of spheres of the radii p with the centres at the points M, that is: S M is the "generalized" sphere of the radius p having its centre at the point M.
Statement 6 [1].The following presentation is valid: ( ) ( ) Corollary.For the following take place: ( ) . [1].The following relation is valid: and the next one is valid: the following is valid:

The Sum of Facets in B n
A facet, or sub-cube, or interval in n B is the set of points satisfying the following condition [5] [6]: where (≤) is a coordinate-wise partial order relation in n B : , where ( ) In other words, an interval can be given by a word of the length n in the alphabet { } 0,1, c , the letters of which are ordered linearly: 0 1 c < < .Indeed, if: of the interval J is built in the following way.
Let ( ) ( ) If ( ) ( ) is the code of the interval J, then all points of the interval J are obtained from the code ( ) J λ by replacing the letters in an arbitrary way by zeros or units.Let ( ) λ be the numbers of letters 1 and c, respectively included in ( ) ( ) If the operation "⋆" is introduced in the alphabet A by the following Caley table [7]: then the sum of the intervals J of the system defined above as a sum of subsets is the interval the code of which is calculated by the codes of items (addends) using the above Caley table.
Thus, the distance between the intervals 1 J and 2 J is the number of occurrence of letters 1 in the code of their sum.

{ }
, x B x p = ∈ = be the p-th layer of an-dimensional cube or sphere of the radius p and the centre at zero [9] [10].
By definition n n p q B B + is the sum of layers in n B , consisting of the union of sums of the points one of which has the weight and the second has the weight q.It is clear that the symmetrical group n S operates on each layer in the following manner: if Hence, g permutes the coordinates of the point, leaving its Hamming weight unchanged.
At the same time the relation ( ) ( ) ( ) B is a transitive set or an orbit of operation of the group n S on the cube n B .Let p g a − = , p g b + =.Statement 9 [1].The following formula is valid: For not large values of the layer the following table of addition is valid: Note that Formula (3) can be rewritten for any number of terms, using the above-mentioned property of distributivity.
Indeed, using (3), we get: As each layer n p B is a sphere of the radius p and the centre at zero point, then all the preceding formulae are rules of 'sphere' addition.

Sum of Subsets in B n
If we take subspaces in n B as terms of the sum X Y + , we will get a well-known object.Indeed, if , X Y is a subspace in n B , then ( ) + is a subspace, too, and we have: Thus, "theory of addition of subspaces" being a well-developed part of linear algebra, makes possible to answer many questions concerning the subject problem.

Sum of Spheres in B n
The k-dimensional interval we denote by k J .According to statement 6, we have: ( ) is the union of all spheres of the radii t with centres at the points in the interval k J , or: ( ) ( ) For the cardinality of the set ( ) is the cardinality of the sphere of the radius 1 t in n k B − .
, for any point the following is valid: n k t β β − ≤ Indeed, in this casebelongs to the sphere of the radius t with the centre at ( ) ,

Sum of a Layer and an Interval in B n
Analogous to the preceding statement and corollary we get the sum of the sets n k t

B J
+ .Statement 11.The following relation is valid: .
Corollary.The cardinality of the set n k t

B J
+ is calculated as follows: ( )

Sum of a Sphere and a Layer in B n
Statement 12.The following is valid: ( , min , , max 0, 1. l p q n l p q = + = − − Proof.We have from statements 9 and 6:

Equation in Sets
Let ( ) B + be the monoid of all subsets with operation of addition (1) in n B as was defined above.This monoid is of certain interest both in classical discrete analysis [8] and for a number of problems connected with theory of information [4].
The 'simplest' equation in sets is as follows: ; consequently, the Hausdorff distance between the sets X and Y: is expressed by the norm of the sum of these solutions.
On the other hand, if: , R X Y is the reciprocal spectrum of the distance between the points of the sets X and Y and: is the spectrum of the distance between the points of the set X, or rather, the spectrum of X.
Thus, the set X X + describes, to a considerable extent, the set of distances between the points of X or the spectrum of X.
In an additive channel of communication [4] the class of equivalence has one to one presentation by transitive sets of certain 'generating' channels.The problem is to order these transitive sets through cardinalities of 'generating' channels.We need the following numerical parameters, which depend on solutions of Equation ( 4) and on the right hand side of A. Let ( ) M A is explained by the fact that the equation X X A + = can sometimes have no solution (for instance, for 3, 5; or for 0 A ∉ ), though the equation X X A + = always has a solution.
Then, for the minimal and maximal cardinality set X Y ∪ , where ( ) ( ) we get respective boundary values, which make possible to narrow the region ( ) N A , i.e. the region of the set of solutions of Equation ( 4) (we shall see this below).
It is not hard to prove that: As every solution ( ) , X X of the equation X X A + = is a solution for ( 4), then we present the following useful statement which makes possible to obtain solutions of the equation X X Y + = from solutions of the Equation ( 4), under certain limitations.
and ( ) . Proof.It follows from ( 5) that it is sufficient to prove for (a) that: is a solution of the equation X Y A + =, for which: On the other hand, it follows from Statement 13 that ( ) Taking into account this and ( 6), we get:

( ) ( ). m A m A ≥
The proof for the case (b) is analogical.Statement 15.The following estimations are valid: ( ) ( ) Proof.Items 1 and 2 of this statement were proved in [1], and we prove only item 3.
Necessity.We assume that: ( ) ( ) and that the pair ( ) According to the statement, we have that the set ( ) We consider a Boolean matrix ( ), X Y ∪ in its rows.We denote by i k the number of units in the i-thcolumn of this matrix.As A is a subspace, then the following equality is true: , ( ) For 2 i s ≥ and 1 2 i k s − − ≥ this equation has no solution for every k.Consequently, 1 Now it is easy to find the solution of Equation ( 9): , , , k e e e be the basis for the space k A .We consider the following sets: { } ( ) The statement is proved. Examples.

Let
We have: Consequently, the upper limit is not reached in this example.
, then the solution of the equation X X A + = is an equidistant code with a distance between any two points equal k, and ( ) At the same time ( ) 3 111 , 2 1 0 0 , 2 0 1 0 , 2 0 0 1 It is clear that by definition 1 2 1 2 , in which the cardinality of the multiset is the sum of the multiplicities of its members.
In particular, the following expression is valid: { } where ( ) ( )

, ;
; a r x y x y z r . min , n a r a r x y x y z p g r r From this, taking into account Statement 9, we get Formula (12).

Corollary. For g p n ≤ ≤
we have: is the multiplicity of the members of ( ) is the multiplicity of the members .
n m x B v v ∈ + + Corollary.For

2 .
The lower limit is achieved if there exists

1 .. 2 .
First we build the multiset C X X = + Then we choose all the pairs in C having the multiplicity m.
can choose n B for X, and any subset of n B for Y. 2. If A is a subspace of n B , then A A A + = and, therefore, Equation (4) has the solution .
If we keep to these examples, then we can assume that there exists some monotonous dependence of the function ( ) m A on the cardinality A. But one can manage to find the possible connection between the right hand side of Equation (4) and the function ( ) m A for the case if A is the halfspace.Corollary.For the halfspace 1 2

∪
where C is an arbitrary subset in n B and C is the multiplicity of the constituent of the members of X Y is called additive if for any , n G B ⊆ and the action of this subgroup on the family 2 n B : { } n B

sphere of the radius t with the centre at the point
 Proof.Let z be any member of the multiset .