An atom of a regular language L with n (left) quotients is a non-empty intersection of uncomplemented or complemented quotients of L, where each of the n quotients appears in a term of the intersection. The quotient complexity of L, which is the same as the state complexity of L, is the number of quotients of L. We prove that, for any language L with quotient complexity n, the quotient complexi...

Let G $G$ be a finite group and recall that the Frattini subgroup Frat ( ) ${\rm Frat}(G)$ is intersection of all maximal subgroups . In this paper, we investigate number , denoted α $\alpha (G)$ which minimal whose coincides with earlier work, studied in special case where simple here extend analysis to almost groups. particular, prove ⩽ 4 (G) \leqslant 4$ for every best possible. We also esta...

If m(n,l) denotes the maximum number of subsets of an n-element set such that the intersection of any two of them has cardinality divisible by l, then a trivial construction shows that m(n,l) ≥ 2 [n / l]. For l = 2, this was known to be essentially best possible. For l ≥ 3, we show by construction that m(n,l) 2 − [n / l] grows exponentially in n, and we provide upper bounds.

We consider mixed integer linear programs where free integer variables are expressed in terms of nonnegative continuous variables. When this model only has two integer variables, Dey and Louveaux characterized the intersection cuts that have infinite split rank. We show that, for any number of integer variables, the split rank of an intersection cut generated from a rational lattice-free polyto...

We consider a vertex model in d dimensions characterized by lines which run in a preferred direction. We show that this vertex model is soluble if the weights of vertices with intersecting lines are given by a free-fermion condition, and that a fugacity −1 is associated to each loop of lines. The solution is obtained by mapping the model into a dimer problem and by evaluating a Pfaffian. We als...

In this paper, our purpose is twofold. Firstly, the tensor andresiduum operations on $L-$nested systems are introduced under thecondition of complete residuated lattice. Then we show that$L-$nested systems form a complete residuated lattice, which isprecisely the classical isomorphic object of complete residuatedpower set lattice. Thus the new representation theorem of$L-$subsets on complete re...

The intersection graph $mathbb{Int}(A)$ of an $S$-act $A$ over a semigroup $S$ is an undirected simple graph whose vertices are non-trivial subacts of $A$, and two distinct vertices are adjacent if and only if they have a non-empty intersection. In this paper, we study some graph-theoretic properties of $mathbb{Int}(A)$ in connection to some algebraic properties of $A$. It is proved that the fi...

Let X be an n-set and L a set of nonnegative integers. F , a set of subsets of X, is said to be an L -intersection family if and only if for all E 6= F ∈ F , |E ∩F | ∈ L. A special case of a conjecture of Frankl and Füredi [4] states that if L = {1, 2, . . . , k}, k a positive integer, then |F| ≤ ∑ki=0 (n−1 i ). Here |F| denotes the number of elements in F . Recently Ramanan proved this conject...

obtain a new for the of a-(u, A) design the block intersection designs are eliminated by an ad hoc coding theoretic argument. A 2-(v, k, A) design 93 is said to be quasi-symmetric if there are two block intersection sizes s1 and s2. The parameters of the complementary design !3* are related to the parameters of 93 as follows: Here Ai denotes the number of blocks through a given i points (and A ...

The celebrated Erdös, Faber and Lovász Conjecture may be stated as follows: Any linear hypergraph on v points has chromatic index at most v. We will introduce the linear intersection number of a graph, and use this number to give an alternative formulation of the Erdös, Faber, Lovász conjecture. Finally, first results about the linear intersection number will be proved. For example, the definit...