On lattices and their ideal lattices, and posets and their ideal posets

For P a poset or lattice, let Id(P) denote the poset, respectively, lattice, of upward directed downsets in P, including the empty set, and let id(P)=Id(P)-\{\emptyset\}. This note obtains various results to the effect that Id(P) is always, and id(P) often, "essentially larger" than P. In the first vein, we find that a poset P admits no "<"-respecting map (and so in particular, no one-to-one isotone map) from Id(P) into P, and, going the other way, that an upper semilattice S admits no semilattice homomorphism from any subsemilattice of itself onto Id(S). The slightly smaller object id(P) is known to be isomorphic to P if and only if P has ascending chain condition. This result is strengthened to say that the only posets P_0 such that for every natural number n there exists a poset P_n with id^n(P_n)\cong P_0 are those having ascending chain condition. On the other hand, a wide class of cases is noted here where id(P) is embeddable in P. Counterexamples are given to many variants of the results proved.Comment: 8 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be updated more frequently than arXiv copy. After publication, updates, errata, etc. may be noted at that pag

More Abelian groups with free duals

In answer to a question of A. Blass, J. Irwin and G. Schlitt, a subgroup G of the additive group Z^{\omega} is constructed whose dual, Hom(G,Z), is free abelian of rank 2^{\aleph_0}. The question of whether Z^{\omega} has subgroups whose duals are free of still larger rank is discussed, and some further classes of subgroups of Z^{\omega} are noted.Comment: 9 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be updated more frequently than arXiv cop

Thoughts on Eggert's Conjecture

Eggert's Conjecture says that if R is a finite-dimensional nilpotent commutative algebra over a perfect field F of characteristic p, and R^{(p)} is the image of the p-th power map on R, then dim_F R \geq p dim_F R^{(p)}. Whether this very elementary statement is true is not known. We examine heuristic evidence for this conjecture, versions of the conjecture that are not limited to positive characteristic and/or to commutative R, consequences the conjecture would have for semigroups, and examples that give equality in the conjectured inequality. We pose several related questions, and briefly survey the literature on the subject.Comment: 12 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be updated more frequently than arXiv copy. A few misstatements in the first version have been corrected, and the wording improved in place

On monoids, 2-firs, and semifirs

Several authors have studied the question of when the monoid ring DM of a monoid M over a ring D is a right and/or left fir (free ideal ring), a semifir, or a 2-fir (definitions recalled in section 1). It is known that for M nontrivial, a necessary condition for any of these properties to hold is that D be a division ring. Under that assumption, necessary and sufficient conditions on M are known for DM to be a right or left fir, and various conditions on M have been proved necessary or sufficient for DM to be a 2-fir or semifir. A sufficient condition for DM to be a semifir is that M be a direct limit of monoids which are free products of free monoids and free groups. W.Dicks has conjectured that this is also necessary. However F.Ced\'o has given an example of a monoid M which is not such a direct limit, but satisfies the known necessary conditions for DM to be a semifir. It is an open question whether for this M, the rings DM are semifirs. We note some reformulations of the known necessary conditions for DM to be a 2-fir or a semifir, motivate Ced\'o's construction and a variant, and recover Ced\'o's results for both constructions. Any homomorphism from a monoid M into \Z induces a \Z-grading on DM, and we show that for the two monoids in question, the rings DM are "homogeneous semifirs" with respect to all such nontrivial \Z-gradings; i.e., have (roughly) the property that every finitely generated homogeneous one-sided ideal is free. If M is a monoid such that DM is an n-fir, and N a "well-behaved" submonoid of M, we obtain results on DN. Using these, we show that for M a monoid such that DM is a 2-fir, mutual commutativity is an equivalence relation on nonidentity elements of M, and each equivalence class, together with the identity element, is a directed union of infinite cyclic groups or infinite cyclic monoids. Several open questions are noted.Comment: 28 pages. To appear, Semigroup Forum. Some clarifications and corrections from previous versio

Double-V block fingers with cruciform recess

In a robot having a gripper including a pair of fingers and a drive motor for driving the fingers toward and away from one another while the fingers remain parallel to each other, the fingers consist of finger pads, which interface with a handle on an object to be grasped, and a shank, which attaches the fingers to the robot gripper. The double-V finger has two orthogonal V-grooves forming in the center of the finger pads and recessed cruciform. The double-V finger is used with a handle on the object to be grasped which is the negative of the finger pads. The handle face consists of V-shaped pads capped with a rectangular cruciform. As the gripper is brought into place near the handle, the finger pads are lined up facing the handle pads. When the finger pad and the handle pad are in proper alignment, the rectangular ridges on the handle fall inside the rectangular grooves on the finger, and the grip is complete

A reservoir of test items for junior high school American history.

Thesis (Ed.M.)--Boston Universit

Continuity of homomorphisms on pro-nilpotent algebras

Let V be a variety of not necessarily associative algebras, and A an inverse limit of nilpotent algebras A_i\in V, such that some finitely generated subalgebra S \subseteq A is dense in A under the inverse limit of the discrete topologies on the A_i. A sufficient condition on V is obtained for all algebra homomorphisms from A to finite-dimensional algebras B to be continuous; in other words, for the kernels of all such homomorphisms to be open ideals. This condition is satisfied, in particular, if V is the variety of associative, Lie, or Jordan algebras. Examples are given showing the need for our hypotheses, and some open questions are noted.Comment: Apologies; in submitting version 2, I didn't realize I had to delete version 1; so the result was a mess. Here is the proper revised version. 23 pages, to appear, Ill.J.Math. Main changes in Aug.2010 revision: Re-formatted to fit journal-sized page. New Section 8 added, on question of when subalgebras of finite codimension must be open. Proofs of Lemma 5 and of Lemma 6(iii) shortene

The Underground Economy: An Overview and Estimates for Cyprus

The paper begins by describing three important macroeconomic approaches to the measurement of the underground economy. Estimates of the size of the underground economy in Cyprus are then discussed. The estimates are derived using a method first applied by Tanzi to data for the United States. Using annual times series data for the period 1960-1990 the size of the underground economy in Cyprus is estimated to be, approximately, between 3% and 10% of GNP.