This paper is devoted to settling the following problem on (infinite, partially) ordered sets: Is there always a partition (2-coloring) of an ordered set X so that all nontrivial maximal chains ofX meet both classes (receive both colors)? We show this is true for all countable ordered sets and provide counterexamples of cardinality H3. Variants of the problem are also considered and open proble...

We show that the maximal linear extension theorem for well partial orders is equivalent over RCA0 to ATR0. Analogously, the maximal chain theorem for well partial orders is equivalent to ATR0 over RCA0.

It is known that a graded lattice of rank n is supersolvable if and only if it has an EL-labelling where the labels along any maximal chain are exactly the numbers 1, 2, . . . , n without repetition. These labellings are called Sn EL-labellings, and having such a labelling is also equivalent to possessing a maximal chain of left modular elements. In the case of an ungraded lattice, there is a n...

Scorpion venom is the richest source of peptide toxins with high levels of specific interactions with different ion-channel membrane proteins. The present study involved the amplification and sequencing of a 310-bp cDNA fragment encoding a beta-like neurotoxin active on sodium ion-channel from the venom glands of scorpion Androctonus crassicauda belonging to the Buthidae family using r...

the main objective of this study is to swing krull intersection theorem in primary decomposition of rings and modules to the primary decomposition of soft rings and soft modules. to fulfill this aim several notions like soft prime ideals, soft maximal ideals, soft primary ideals, and soft radical ideals are introduced for a soft ring over a given unitary commutative ring. consequently, the p...

We characterize supersolvable lattices in terms of a certain modular type relation. McNamara and Thomas earlier characterized this class as those graded having maximal chain that consists left-modular elements. Our characterization replaces the condition gradedness with second modularity on

Let L be a finite distributive lattice. Let Sub0(L) be the lattice {S | S is a sublattice of L} ∪ {∅} and let ∗[Sub0(L)] be the length of the shortest maximal chain in Sub0(L). It is proved that if K and L are non-trivial finite distributive lattices, then ∗[Sub0(K × L)] = ∗[Sub0(K)] + ∗[Sub0(L)]. A conjecture from the 1984 Banff Conference on Graphs and Order is thus proved. 1 Motivation Let L...

We investigated maximal Prym varieties on finite fields by attaining their upper bounds on the number of rational points. This concept gave us a motivation for defining a generalized definition of maximal curves i.e. maximal morphisms. By MAGMA, we give some non-trivial examples of maximal morphisms that results in non-trivial examples of maximal Prym varieties.