Rafael Diaz, Mariolys Rivas
We define Boolean algebras in the linear context and study its symmetric powers. We give explicit formulae for products in symmetric Boolean algebras of various dimensions. We formulate symmetric forms of the inclusion-exclusion principle.
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The random ordered graph is a semi-retract of the canonically ordered atomless Boolean algebra
本文证明随机有序图是规范有序无原子布尔代数的半收缩，回答了Bartošová和Scow的公开问题。
This work presents a study of perturbations of symmetric Boolean functions. In particular, it establishes a connection between exponential sums of these perturbations and Diophantine equations of the form $$ \sum_{l=0}^n \binom{n}{l} x_l=0,$$ where $x_j$ belongs to some fixed bounded subset $Γ$ of $\mathbb{Z}$. The concepts of trivially balanced symmetric Boolean function and sporadic balanced Boolean function are extended to this type of perturbations. An observation made by Canteaut and Videau for symmetric Boolean functions of fixed degree is extended. To be specific, it is proved that, exc…
Oscillating subalgebras of the atomless countable Boolean algebra
该研究证明了具有3个原子的布尔代数在可数无原子布尔代数中的大拉姆齐度为无穷，推进了拉姆齐理论在布尔代数领域的相关研究
R. Canfield has conjectured that for all subgroups G of the automorphism group of the Boolean lattice B(n) (which can be regarded as the symmetric group S(n)) the quotient order B(n)/G is a symmetric chain order. We provide a straightforward proof of a generalization of a result of K. K. Jordan: namely, B(n)/G is an SCO whenever G is generated by powers of disjoint cycles. The symmetric chain decompositions of Greene and Kleitman provide the basis for partitions of these quotients.
The finite-degree Zariski (Z-) closure is a classical algebraic object, that has found a key place in several applications of the polynomial method in combinatorics. In this work, we characterize the finite-degree Z-closures of a subclass of symmetric sets (subsets that are invariant under permutations of coordinates) of the Boolean cube, in positive characteristic. Our results subsume multiple statements on finite-degree Z-closures that have found applications in extremal combinatorial problems, for instance, pertaining to set systems (Hegedűs, Stud. Sci. Math. Hung. 2010; Hegedűs, arXiv 2021…
On Symmetric Pseudo-Boolean Functions: Factorization, Kernels and Applications
该研究证明对称伪布尔函数可等价表示为幂级数或因式分解，其核对应至少一个n维仿射超平面，并将结果应用于自旋玻璃、量子信息与张量网络领域的相关函数分析。
Nonlinearity of rotation symmetric Boolean functions is an important topic on cryptography algorithm. Let $e\ge 1$ be any given integer. In this paper, we investigate the following question: Is the nonlinearity of the quartic rotation symmetric Boolean function generated by the monomial $x_0x_ex_{2e}x_{3e}$ equal to its weight? We introduce some new simple sub-functions and develop new technique to get several recursive formulas. Then we use these recursive formulas to show that the nonlinearity of the quartic rotation symmetric Boolean function generated by the monomial $x_0x_ex_{2e}x_{3e}$ i…
We apply the Dual Ramsey Theorem of Graham and Rothschild to prove the Ramsey property for classes of finite Boolean algebras with distinguished ideals. This allows us to compute the universal minimal flow of the group of automorphisms of $\mathcal{P}(ω_1)/\rm{fin},$ should it be isomorphic to $\mathcal{P}(ω)/\rm{fin}$ or not, and of other quotients of power set algebras. Taking Fraïssé limits of these classes, we can compute universal minimal flows of groups of homeomorphisms of the Cantor set fixing some closed subsets.
The quotient of a Boolean algebra by a cyclic group is proven to have a symmetric chain decomposition. This generalizes earlier work of Griggs, Killian and Savage on the case of prime order, giving an explicit construction for any order, prime or composite. The combinatorial map specifying how to proceed downward in a symmetric chain is shown to be a natural cyclic analogue of the $\mathfrak{sl}_2$ lowering operator in the theory of crystal bases.
Let $B$ be a finite Boolean algebra. Let $\mathcal A$ be the partial order of all implication sublattices of $B$. We will compute the Möbius function on $\mathcal A$ in two different ways.
In this paper, we study relationships between the normalized characters of symmetric groups and the Boolean cumulants of Young diagrams. Specifically, we show that each normalized character is a polynomial of twisted Boolean cumulants with coefficients being non-negative integers, and conversely, that, when we expand a Boolean cumulant in terms of normalized characters, the coefficients are again non-negative integers. The main tool is Khovanov's Heisenberg category and the recently established connection of its center to the ring of functions on Young diagrams, which enables one to apply grap…
Sensitivity of Rotation Symmetric Boolean Functions via Binary Necklaces
该研究针对旋转对称布尔函数，借助二元项链推导敏感性公式，明确奇素数变量、单项式函数等的敏感性，为相关函数的敏感性分析提供了精确结果。
Alon and Füredi (European J. Combin., 1993) proved that any family of hyperplanes that covers every point of the Boolean cube $\{0,1\}^n$ except one must contain at least $n$ hyperplanes. We obtain two extensions of this result, in characteristic zero, for hyperplane covers of symmetric sets of the Boolean cube (subsets that are closed under permutations of coordinates), as well as for `polynomial covers' of `weight-determined' sets of `strictly unimodal uniform' (SU$^2$) grids. As a main tool for solving our problems, we give a combinatorial characterization of (finite-degree) Zariski (Z-) cl…
Rotation symmetric Boolean functions are invariant under circular translation of indices. These functions have very rich cryptographic properties and have been used in different cryptosystems. Recently, Thomas Cusick proved that exponential sums of rotation symmetric Boolean functions satisfy homogeneous linear recurrences with integer coefficients. In this work, a generalization of this result is proved over any Galois field. That is, exponential sums over Galois fields of rotation symmetric polynomials satisfy linear recurrences with integer coefficients. In the particular case of $\mathbb{F…
Let G be a finite group acting on the finite set X such that the corresponding (complex) permutation representation is multiplicity free. There is a natural rank and order preserving action of the wreath product G~S_n on the generalized Boolean algebra B_X(n). We explicitly block diagonalize the commutant of this action.
We introduce the antipodal pairs property for probability measures on finite Boolean algebras and prove that conditional versions imply strong forms of log-concavity. We give several applications of this fact, including improvements of some results of Wagner; a new proof of a theorem of Liggett stating that ultra-log-concavity of sequences is preserved by convolutions; and some progress on a well-known log-concavity conjecture of J. Mason.
Sensitivity and Block Sensitivity of Elementary Symmetric Boolean Functions of Arbitrary Degree
本文完全确定了任意次基本对称布尔函数的敏感性、平均敏感性和块敏感性，并分类了其与证书复杂度的三种关系模式，给出了严格层级的充要条件。
In this paper, we investigate permutation rotation-symmetric (shift-invariant) vectorial Boolean functions on $n$ bits that are liftings from Boolean functions on $k$ bits, for $k\leq n$. These functions generalize the well-known map used in the current Keccak hash function, which is generated via the Boolean function on $3$ variables, $x_1+(x_2+1)x_3$. We provide some general constructions, and also study the affine equivalence between rotation-symmetric S-boxes and describe the corresponding relationship between the Boolean function they are associated with.
We define an action of the 0-Hecke algebra of type A on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their (q,t)-analogues introduced by Bergeron and Zabrocki, and to a more general family of noncommutative symmetric functions having parameters associated with paths in binary trees introduced recently by Lascoux, Novelli, and Thibon. We also obtain multivariate quasisymmetric function identities, which specialize to re…
Writing the values of Krawtchouk polynomials as matrices, we consider weighted partial sums along columns. For the general case, we find an identity that, in the symmetric case yields a formula for such partial sums. Complete sums of squares along columns involve "Super Catalan" numbers. We look as well for particular values (matrix entries) involving the Catalan numbers. Properties considered and developed in this work are applied to calculations of various dimensions that describe the structure of some *-algebras over the Boolean lattice based on inclusion/superset relations expressed algebr…
The theory of cumulants is revisited in the "Rota way", that is, by following a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants are considered as infinitesimal characters over a particular combinatorial Hopf algebra. The latter is neither commutative nor cocommutative, and has an underlying unshuffle bialgebra structure which gives rise to a shuffle product on its graded dual. The moment-cumulant relations are encoded in terms of shuffle and half-shuffle exponentials. It is then shown how to express concisely monotone, free, and boolean cumulants in terms of each oth…
Rotation symmetric Boolean functions have important applications in the design of cryptographic algorithms. In this paper, the Conjecture about rotation symmetric Boolean functions (RSBFs) of degree 3 proposed by Cusik and Stănică is proved. As a result, the nonlinearity of such kind of functions is determined.
We determine the dimension of every simple module for the algebra of the monoid of all relations on a finite set (i.e. Boolean matrices). This is in fact the same question as the determination of the dimension of every evaluation of a simple correspondence functor. The method uses the theory of such functors developed in [BT2, BT3], as well as some new ingredients in the theory of finite lattices.
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