In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear orders has a five element basis. In fact such a basis follows from the Proper Forcing Axiom, a strong form of the Baire Category Theorem. The elements are X , ω1,...

There are several graphs associated with finite groups which are currently of interest in the field. This subject of this project are graphs related to the element orders of a finite group. The best known such graph is the prime graph Γ(G) of the group G and is defined as follows. The vertices are the prime numbers dividing the order (=number of elements) of the group, and two such primes are l...

Well-founded orders are the opposite of noetherian orders: every nonempty subset contains at least one minimal element. And a set is well-ordered when it is totally ordered by a wellfounded order: every nonempty set contains exactly one minimal element. Sofar, noetherian induction has been the most powerful way of proving properties inductively; it is indeed the most general one in the precise ...

suppose that  is a finite group. then the set of all prime divisors of  is denoted by  and the set of element orders of  is denoted by . suppose that . then the number of elements of order  in  is denoted by  and the sizes of the set of elements with the same order is denoted by ; that is, . in this paper, we prove that if  is a group such that , where , then . here  denotes the family of suzuk...

We study finite partial orders which have a chain such that every element of the order either belongs to this chain or has all its covers in this chain. We show that such orders are exactly the orders being both interval orders and truncated lattices. We prove that their jump number is polynomially tractable and that their dimension is unbounded. We also show that every order admits a visibilit...

We study two kinds of segment orders, using definitions first proposed by Farhad Shahrokhi. Although the two kinds of segment orders appear to be quite different, we prove several results suggesting that the are very much the same. For example, we show that the following classes belong to both kinds of segment orders: (1) all posets having dimension at most 3; (2) interval orders; and for n ≥ 3...

Let r be a fixed positive integer. It is shown that, given any partial orders >1, . . ., >r on the same n-element set P , there exist disjoint subsets A,B ⊂ P , each with at least n1−o(1) elements, such that one of the following two conditions is satisfied: (1) there is an i (1 ≤ i ≤ r) such that every element of A is larger than any element of B in the partial order >i, or (2) no element of A ...

Vendor performance plays a key role in price, quality, timely delivery and service in achieving the goals of the supply chain. Given this, today's supplier performance has become a critical element in the success or failure of a company. In this study, an idealized planning model with multi-criteria decision-making approach with network analysis tool based on the expansion of fuzzy quality func...

Abstract Let denote the maximal number of different primes that may occur in order a finite solvable group G , all elements which have orders divisible by at most n distinct primes. We show for . As an application, we improve on recent bound Hung and Yang arbitrary groups.