Serial rings and tiled orders of width two

We construct Artinian serial rings and tiled orders of width two with maximal finite global dimension.     

On the finitistic dimension conjecture II: Related to finite global dimension

In this paper, we study the finitistic dimensions of artin algebras by establishing a relationship between the global dimensions of the given algebras, on the one hand, and the finitistic dimensions of their subalgebras, on the other hand. This is a continuation of the project in [J. Pure Appl. Algebra 193 (2004) 287-305]. For an artin algebra A we denote by gl.dim(A), fin.dim(A) and rep.dim(A) the global dimension, finitistic dimension and representation dimension of A, respectively. The Jacobson radical of A is denoted by rad(A). The main results in the paper are as follows: Let B be a subalgebra of an artin algebra A such that rad(B) is a left ideal in A. Then (1) if gl.dim(A)?4 and rad(A)=rad(B)A, then fin.dim(B)<∞. (2) If rep.dim(A)?3, then fin.dim(B)<∞. The results are applied to pullbacks of algebras over semi-simple algebras. Moreover, we have also the following dual statement: (3) Let ?:B?A be a surjective homomorphism between two algebras B and A. Suppose that the kernel of ? is contained in the socle of the right B-module B_B. If gl.dim(A)?4, or rep.dim(A)?3, then fin.dim(B)<∞. Finally, we provide a class of algebras with representation dimension at most three: (4) If A is stably hereditary and rad(B) is an ideal in A, then rep.dim(B)?3.     

Random orders of dimension 2

A relationship is established between (partially) ordered sets of dimension 2 chosen randomly on a labelled set, chosen randomly by isomorphism type, or generated by pairs of random linear orderings. As a consequence we are able to determine the limiting probability (in each of the above sample spaces) that a two-dimensional order is rigid, is uniquely realizable, or has uniquely orientable comparability graph; all these probabilities lie strictly between 0 and 1. Finally, we show that the number of 2-dimensional (partial) orderings of a labelled n-element set is .On leave from Emory University, Atlanta, GA. Research at Emory supported by ONR grant N00014 85-K-0769.     

The dimension of cycle-free orders

The existence of a four-dimensional cycle-free order is proved. This answers a question of Ma and Spinrad. Two similar problems are also discussed.Research partially supported by Office of Naval Research grant N00014-90-J-1206Research partially supported by the National Science Foundation under grant DMS     

Fractional dimension of partial orders

Given a partially ordered setP=(X, ), a collection of linear extensions {L ₁,L ₂,...,L _r } is arealizer if, for every incomparable pair of elementsx andy, we havex<y in someL _i (andy<x in someL _j ). For a positive integerk, we call a multiset {L ₁,L ₂,...,L _t } ak-fold realizer if for every incomparable pairx andy we havex<y in at leastk of theL _i 's. Lett(k) be the size of a smallestk-fold realizer ofP; we define thefractional dimension ofP, denoted fdim(P), to be the limit oft(k)/k ask. We prove various results about the fractional dimension of a poset.Research supported in part by the Office of Naval Research.     

Global dimension two orders are quasi-hereditary

Motivated by results of Cline, Parshall and Scott on quasi-hereditary algebras, in [8] the concept of a quasi-hereditary order is introduced in integral representation theory. In this note we show that the results of Dlab and Ringel on quasi-hereditary semiprimary rings and hereditary artinian rings presented in [6] have integral analogues in the theory of orders. In particular, we prove as our main result the followingTheorem: An order of global dimension at most two over a complete Dedekind domain R in a separable algebra over the quotient field of R is quasi-hereditary.     

Infinite-dimensional quasigroups of finite orders

Let Σ be a finite set of cardinality k > 0, let $\mathbb{A}$ be a finite or infinite set of indices, and let $\mathcal{F} \subseteq \Sigma ^\mathbb{A}$ be a subset consisting of finitely supported families. A function $f:\Sigma ^\mathbb{A} \to \Sigma$ is referred to as an $\mathbb{A}$ -quasigroup (if $\left| \mathbb{A} \right| = n$ , then an n-ary quasigroup) of order k if $f\left( {\bar y} \right) \ne f\left( {\bar z} \right)$ for any ordered families $\bar y$ and $\bar z$ that differ at exactly one position. It is proved that an $\mathbb{A}$ -quasigroup f of order 4 is reducible (representable as a superposition) or semilinear on every coset of $\mathcal{F}$ . It is shown that the quasigroups defined on Σ^?, where ? are positive integers, generate Lebesgue nonmeasurable subsets of the interval [0, 1].     

On the Boolean dimension of spherical orders

It is well known that if a planar order P is bounded, i.e. has only one minimum and one maximum, then the dimension of P (LD(P)) is at most 2, and if we remove the restriction that P has only one maximum then LD(P)3. However, the dimension of a bounded order drawn on the sphere can be arbitrarily large.The Boolean dimension BD(P) of a poset P is the minimum number of linear orders such that the order relation of P can be written as some Boolean combination of the linear orders. We show that the Boolean dimension of bounded spherical orders is never greater than 4, and is not greater than 5 in the case the poset has more than one maximal element, but only one minimum. These results are obtained by a characterization of spherical orders in terms of containment between circular arcs.Part of this work was carried out while both authors were visiting the Department of Applied Mathematics (KAM) of Charles University, Prague. The authors acknowledge support from the EU HCM project DONET.     

Estimates of global dimension

In this note we show that for a *ⁿ-module, in particular, an almost n-tilting module, P over a ring R with A = End_R P such that P _A has finite flat dimension, the upper bound of the global dimension of A can be estimated by the global dimension of R and hence generalize the corresponding results in tilting theory and the ones in the theory of *-modules. As an application, we show that for a finitely generated projective module over a VN regular ring R, the global dimension of its endomorphism ring is not more than the global dimension of R.     

Two-dimensional orders of finite representation type

Maximal orders of finite representation type over complete local rings of dimension two and of characteristic zero are classified. This completes the classification begun in [1] for the case that R is a power series ring in two variables.     

Degrees of freedom versus dimension for containment orders

Given a family of sets L, where the sets in L admit k degrees of freedom, we prove that not all (k+1)-dimensional posets are containment posets of sets in L. Our results depend on the following enumerative result of independent interest: Let P(n, k) denote the number of partially ordered sets on n labeled elements of dimension k. We show that log P(n, k)nk log n where k is fixed and n is large.Research supported in part by Allon Fellowship and by a grant from Bat Sheva de Rothschild Foundation.Research supported in part by the Office of Naval Research, contract number N00014-85-K0622.