A partially ordered set (poset) is planar if it has a planar Hasse diagram. The dimension of a bounded planar poset is at most two. We show that the dimension of a planar poset having a greatest lower bound is at most three. We also construct four-dimensional planar posets, but no planar poset with dimension larger than four is known. A poset is called a tree if its Hasse diagram is a tree in t...

Let U0, U1 be two normal measures on κ. We say that U0 is in the Mitchell ordering less than U1, U0 C U1, if U0 ∈ Ult(V,U1). The relation is well-known to be transitive and well-founded. It has been an open problem to find a model where C embeds the four-element poset . We find a generic extension where all well-founded posets are embeddable. Hence there is no structural restriction on the Mitc...

The poset retraction problem for a poset P is whether a given poset Q containing P as a subposet admits a retraction onto P, that is, whether there is a homomorphism from Q onto P which fixes every element of P. We study this problem for finite series-parallel posets P. We present equivalent combinatorial, algebraic, and topological charaterisations of posets for which the problem is tractable,...

ibn-khaldun,s social change theory has been often classified as cyclic theoty of social changes that has three stages includes in generation, growth and progress and ultimately regressive and decadence. but, what is inferred by rethink in text of ibn-khaldun,s moghaddame, are two different courses,as interrelated to each other, “civilizational evolution” and “cyclic changes of state.avoiding ma...