A 1984 problem of S. Z. Ditor asks whether there exists a lattice of cardinality א2, with zero, in which every principal ideal is finite and every element has at most three lower covers. We prove that the existence of such a lattice follows from either one of two axioms that are known to be independent of ZFC, namely (1) Martin’s Axiom restricted to collections of א1 dense subsets in posets of ...

This study was conducted to investigate the effect of meaning-focused versus form-focused input-oriented and output-oriented task-based instruction on elementary level Iranian EFL Learners’ vocabulary comprehension and recall. For this purpose, a sample of 120 male students from a private school in Tehran was selected through convenience sampling and based on availability. The participants were...

 In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $|J(L)setminus A(L)| leq 3$, where $J(L)$ and $A(L)$ are the set of join-irreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices ha...

convictions are strong that multimedia courseware holds great promise for enhancing learning efficiency among learners. this study thus aimed at comparing the effects of the multimedia programs introducing 100 vocabulary items through pictures and streaming video with those of teacher instruction on increasing efl learners’ vocabulary repertoire. a total of 120 subjects were identified as inter...

We consider the trade-offs in the choice between depth (a narrow high quality position) and breadth (a wide low quality range). In particular, the extent of depth or breadth in a market can be non-monotonic in the strength of competition.