Related papers: Reaping Numbers of Boolean Algebras

In this paper we prove that every recursively presented Lie algebra over a field which is a finite extention of its simple subfield can be embedded in a recursively presented Lie algebra defined by relations which are equalities of…

Euler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$, \begin{align*}…

We establish some similarities/analogies between uncountable cardinals or powersets and the class $V$ of all sets. They concern mainly the Boolean algebras ${\cal P}(\kappa)$, for a regular cardinal $\kappa$, and ${\cal C}(V)$ (the class of…

Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $0 \leq k \leq n$, let ${[n] \choose \leq…

We prove some general recursions for the numbers of representations of positive integers as a sum x+y, x in X, y in Y, where X,Y are increasing sequences. In particular, we obtain recursions for the number of the Goldbach, Lemoine-Levy,…

Let $A$ be an algebra over any field. We do not assume that $A$ has an identity. The \emph{multiplier algebra} $M(A)$ is a unital algebra associated to $A$. If we require the product in $A$ to be non-degenerate (as a bilinear form), the…

We study the structure of sets $S\subseteq\{0, 1\}^n$ with small sensitivity. The well-known Simon's lemma says that any $S\subseteq\{0, 1\}^n$ of sensitivity $s$ must be of size at least $2^{n-s}$. This result has been useful for proving…

Consider the recursive relation generating a new positive integer $n_{\ell +1}$ from the positive integer $n_{\ell }$ according to the following simple rules: if the integer $n_{\ell }$ is odd, $n_{\ell +1}=3n_{\ell }+1$; if the integer…

There exists a set $A$ of positive integers such that the number of representations of a large positive integer $m$ as a sum of two elements of $A$ grows with a lower bound of order $\log m$, but for which there is no subset $D$ of $A$…

A book $B_n$ is a graph which consists of $n$ triangles sharing a common edge. In 1978, Rousseau and Sheehan conjectured that the Ramsey number satisfies $r(B_m,B_n)\le 2(m+n)+c$ for some constant $c>0$. In this paper, we obtain that…

Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_S(n)$ denote the number of solutions of the equation $n=s+s'$, $s, s'\in S$, $s<s'$. In this paper, we determine the…

An algebraic structure with two constants and one ternary operation, which is not completely commutative, is put forward to accommodate ternary Boolean algebras. When the ternary operation is interpreted as Church's conditioned disjunction,…

Let $ B_{\ell}(n)$ denote the number of $\ell$-regular bipartitions of $n.$ In this article, we prove that $ B_{\ell}(n)$ is always almost divisible by $p_i^j$ if $p_i^{2a_i}\geq \ell,$ where $j$ is a fixed positive integer and…

Let $b$ be an integer greater than or equal to $2$. For any integer $n\in \left[b^{\lambda-1}, b^{\lambda}-1\right]$, we denote by $R_\lambda (n)$ the reverse of $n$ in base $b$, obtained by reversing the order of the digits of $n$. We…

We give a definition of some classes of boolean algebras generalizing free boolean algebras; they satisfy a universal property that certain functions extend to homomorphisms. We give a combinatorial property of generating sets of these…

We prove that if $A,B$ are compact subsets of $\mathbb{R}$ such that the upper density of $B$ is positive at every point of $B$, then there is a closed null set $N\subset A$ such that $N+B=A+B$. As a corollary we find that if $A,B\subset…

We consider the problem of partitioning $n$ integers into two subsets of given cardinalities such that the discrepancy, the absolute value of the difference of their sums, is minimized. The integers are i.i.d. random variables chosen…

A skew Bollob\'{a}s system $\mathcal{P}=\{(A_i,B_i):1\leq i\leq m\}$ is a collection of pairs of disjoint subsets of $[n]$ such that $A_i\cap B_j\ne\emptyset$ for any $1\leq i<j\leq m$. Denote by $S_1(a, b)$ or $S_2(a, b)$ the maximum size…

We construct varieties B(r;An) such that a map X -> B(r;An) corresponds to a degree-n \'etale algebra on X equipped with r generating global sections. We then show that when n = 2, i.e., in the quadratic \'etale case, that the singular…

How many endomorphisms does a Boolean algebra have? Can we find Boolean algebras with as few endomorphisms as possible? Of course from any ultrafilter of the Boolean algebra we can define an endomorphism, and we can combine finitely many…