Equalizers and Flatness Properties of Acts,II

In Comm. Algebra 30 (3) (2002), 1475–1498, Bulman-Fleming and Kilp developed various notions of flatness of a right act A_S over a monoid S that are based on the extent to which the functor A_S$\otimes -$ preserves equalizers. In Semigroup Forum 65 (3) (2002), 428–449, Bulman-Fleming discussed in detail one of these notions, annihilator-flatness. The present paper is devoted to two more of these notions, weak equalizer-flatness and strong torsion-freeness. An act A_S is called weakly equalizer-flat if the functor A_S$\otimes -$ almost preserves equalizers of any two homomorphisms into the left act _SS, and strongly torsion-free if this functor almost preserves equalizers of any two homomorphisms from _SS into the Rees factor act _S(S/Sc), where c is any right cancellable element of S. (The adverb almost signifies that the canonical morphism provided by the universal property of equalizers may be only a monomorphism rather than an isomorphism.) From the definitions it is clear that flatness implies weak equalizer-flatness, which in turn implies annihilator-flatness, and it was known already that both of these implications are strict. A monoid is called right absolutely weakly equalizer-flat if all of its right acts are weakly equalizer-flat. In this paper we prove a result which implies that right PP monoids with central idempotents are absolutely weakly equalizer-flat. We also show that for a relatively large class of commutative monoids, right absolute equalizer-flatness and right absolute annihilator-flatness coincide. Finally, we provide examples showing that the implication between strong torsion-freeness and torsion-freeness is strict.     

Principally Weakly and Weakly Coherent Monoids

We shall call a monoid S principally weakly (weakly) left coherent if direct products of nonempty families of principally weakly (weakly) flat right S-acts are principally weakly (weakly) flat. Such monoids have not been studied in general. However, Bulman-Fleming and McDowell proved that a commutative monoid S is (weakly) coherent if and only if the act S ^I is weakly flat for each nonempty set I. In this article we introduce the notion of finite (principal) weak flatness for characterizing (principally) weakly left coherent monoids. Also we investigate monoids over which direct products of acts transfer an arbitrary flatness property to their components.     

On a conjecture of Bulman-Fleming and Gilmour

If S is a monoid, the set S×S equipped with componentwise S-action is called the diagonal act of S and is denoted by D(S). We prove the following theorem: the right S-act S ⁿ (1≠n∈?) is (principally) weakly flat if and only if \(\prod _{i=1}^{n}A_{i}\) is (principally) weakly flat where A _i , 1≤i≤n are (principally) weakly flat right S-acts, if and only if the diagonal act D(S) is (principally) weakly flat. This gives an answer to a conjecture posed by Bulman-Fleming and Gilmour (Semigroup Forum 79:298–314, 2009). Besides, we present a fair characterization of monoids S over which the diagonal act D(S) is (principally) weakly flat and finally, we impose a condition on D(S) in order to make S a left PSF monoid.     

Examples concerning absolute flatness and amalgamation in pomonoids

Flatness properties of acts over monoids have been studied for almost four decades and a substantial literature is now available on the subject. Analogous research dealing with partially ordered monoids acting on posets was begun in the 1980s in two papers by S.M. Fakhruddin, and, after a dormancy period of some 20 years, has recently been rekindled with the appearance of several research articles. In comparing flatness properties of S-acts and S-posets, it has been noted that the imposition of order results in severe restrictions as far as absolute flatness is concerned. For example, whereas every inverse monoid is absolutely flat (meaning all of its left and right acts are flat), even the three-element chain in its natural order, considered as a pomonoid, fails to have this property. It has long been understood that absolutely flat monoids, in particular, inverse monoids, are amalgamation bases in the class of all monoids. The purpose of the present article is to further investigate absolute flatness of pomonoids and to begin to study its connection with amalgamation in that context. T.E. Hall’s results, that amalgamation bases in the class of all monoids have the so-called representation extension property (REP), which in turn implies the right congruence extension property, are first adapted to the ordered context. A detailed study of the compatible orders (of which there are exactly 13) on the three-element chain semilattice U then reveals a wide range of possibilities: exactly four of these orders render U absolutely flat as a pomonoid, two more give it the right order-congruence extension property in every extension (RCEP) (but fail to make it an amalgamation base because of the failure of the ordered analogue of (REP)), and for the remaining seven, even (RCEP) fails.     

Subpullback flat S-posets need not be subequalizer flat

If S is a monoid, a right S-act A _S is a set A, equipped with a “right S-action” A×S→A sending the pair (a,s)∈ A×S to as, that satisfies the conditions (i) a(st)=(as)t and (ii) a1=a for all a∈A and s,t∈S. If, in addition, S is equipped with a compatible partial order and A is a poset, such that the action is monotone (when A×S is equipped with the product order), then A _S is called a right S-poset. Left S-acts and S-posets are defined analogously. For a given S-act (resp. S-poset) a tensor product functor A _S ?? from left S-acts to sets (resp. left S-posets to posets) exists, and A _S is called pullback flat or equalizer flat (resp. subpullback flat or subequalizer flat) if this functor preserves pullbacks or equalizers (resp. subpullbacks or subequalizers). By analogy with the Lazard-Govorov Theorem for R-modules, B. Stenström proved in 1971 that an S-act is isomorphic to a directed colimit of finitely generated free S -acts if and only if it is both pullback flat and equalizer flat. Some 20 years later, the present author showed that, in fact, pullback flatness by itself is sufficient. (A new, more direct proof of that result is contained in the present article.) In 2005, Valdis Laan and the present author obtained a version of the Lazard-Govorov Theorem for S-posets, in which subpullbacks and subequalizers now assume the role previously played by pullbacks and equalizers. The question of whether subpullback flatness implies subequalizer flatness remained unsolved. The present paper provides a negative answer to this question.     

Excursions above high levels by Gaussian random fields

For a two-dimensional, homogeneous, Gaussian random field X(t) and compact, convex S ? R² we show that as u → ∞ the set A_u = {t ∈ S : X(t) ? u} possesses, with probabilityapproaching one, components that are approximately convex. Furthermore, the function X is also approximately concave over A_u. One of the main aims of the paper is, at the cost of losing some detail, to simplify the analytic complexity of previous results about high level excursions of Gaussian fields by judicious use of concepts from integral and differential geometry.     

Construction of spherical 4- and 5-designs

A finite subsetX of thed-dimensional unit sphereS ^d-1 is called a sphericalt-design, if and only if $$\frac{1}{{\left| {S^{d - 1} } \right|}}\int_{S^{d - 1} } {f(x)d\omega (x)} = \frac{1}{{\left| x \right|}}\sum\limits_{x \in X} {f(x)} $$ holds for all polynomialsf(x) =f(x ₁,x ₂,...,x _d ) of degree at mostt. In 1984 Seymour and Zaslavsky proved the existence of sphericalt-designs for anyt andd, but for sufficiently large |X|. Since spherical designs can be used for numerical integration, it is of interest to give explicit constructions. Mimura gave a construction fort = 2,d ∈ ? and |X| ≥n ₂ for somen ₂ ∈ ? (n ₂ is sharp). Here we will give an explicit construction fort = 4 and 5,d ∈ ? and |X| ≥n ₄ for somen ₄ ∈ ?.     

Principal Weak Flatness and Regularity of Diagonal Acts

For a monoid S, the set S × S equipped with the componentwise right S-action is called the diagonal act of S and is denoted by D(S). A monoid S is a left PP (left PSF) monoid if every principal left ideal of S is projective (strongly flat). We shall call a monoid S left P(P) if all principal left ideals of S satisfy condition (P). We shall call a monoid S weakly left P(P) monoid if the equalities as = bs, xb = yb in S imply the existence of r ∈ S such that xar = yar, rs = s. In this article, we prove that a monoid S is left PSF if and only if S is (weakly) left P(P) and D(S) is principally weakly flat. We provide examples showing that the implications left PSF ? left P(P) ? weakly left P(P) are strict. Finally, we investigate regularity of diagonal acts D(S), and we prove that for a right PP monoid S the diagonal act D(S) is regular if and only if every finite product of regular acts is regular. Furthermore, we prove that for a full transformation monoid S = 𝒯 _X , D(S) is regular.     

Graphs whose strong endomorphism monoids are regular

Let X be a graph, S End X be its strong endomorphism monoid. It is proved that S End X is a regular monoid if and only if the canonical strong factor graph U of X contains no proper subgraph which is isomorphic to U. The result generalizes that of U. Knauer about the regularity of strong endomorphism monoids of graphs.     

A note on zip rings

A ring R is called right zip if the right annihilator r _R (S) of a subset S of R is zero, r _R (X)=0 for a finite subset X of S. In this note we will prove that for any u.p.-monoid M a right uniform ring R is right zip if and only if the monoid ring R[M] is right zip.     

Convergence rates in the law of large numbers for random variables on partially ordered sets

Let (A, ≤) be a partially ordered set, {X _α} a collection of i. i. d. random variables, indexed byA. Let \(S_\alpha = \sum _{\beta \leqslant \alpha } X_\beta \) , |α|=card {β∈A, β∈α}. We study the convergence rates ofS _α/|α|. We derive for a large class of partially ordered sets theorems, like the following one: For suitabler, t with 1/2< <r/t≤1:E|X| ^t M (|X| ^t/r )<∞ andEX=μ if and only if $$S_\alpha = \sum _{\beta \leqslant \alpha } X_\beta $$ for all ε>0, where \(M(x) = \sum _{j< x} d(j)\) withd(j)=card {α∈A, |α|=j}.     

Orders in Uniserial Rings

Let A be a ring, and let T(A) and N(A) be the set of all the regular elements of A and the set of all nonregular elements of A, respectively. It is proved that A is a right order in a right uniserial ring if and only if the set of all regular elements of A is a left ideal in the multiplicative semigroup A and for any two elements a ₁ and a ₂ of A, either there exist two elements b ₁ ∈ A and t ₁ ∈ T(A) with a1b₁ = a ₂t₁ or there exist two elements b ₂∈ A and t ₂∈ T(A) with a ₂ b ₂ = a ₁ t ₂. A right distributive ring A is a right order in a right uniserial ring if and only if the set N(A) is a left ideal of A. If A is a right distributive ring such that all its right divisors of zero are contained in the Jacobson radical J(A) of A, then A is a right order in a right uniserial ring.     

Axiomatizability of the class of weakly injective <Emphasis Type="Italic">S</Emphasis>-acts

The concepts of weakly injective, fg-weakly injective, and p-weakly injective S-acts generalize that of injective S-act. We study the monoids S over which the classes of weakly injective, fg-weakly injective, and p-weakly injective S-acts are axiomatizable. We prove that the class of p-weakly injective S-acts over a regular monoid is axiomatizable.     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.     

Stochastic integral representation of some martingales

Let {X_t} be a continuous square integrable martingale. Denote its increasing (natural) process by {A_t}. Let S_t, T_t be the left and right inverses of A_t, respectively. Then for any square integrable martingale {Y_t} defined on {X_t}, Y_t = ∝₀^tψ_sdX_s, R₀ < t < S_∞ where S_∞ = lim_t→∞S_t, R₀ = inf {t: X_t ≠ 0} provided that Y(T(t)) is σ(X(T(s)): s ? t)-measurable. All martingales are assumed to be zero at t = 0. Brownian motion and Poisson processes are considered also.     

Divisibility theory in commutative rings: Bezout monoids

A ubiquitous class of lattice ordered semigroups introduced by Bosbach, which we call Bezout monoids, seems to be the appropriate structure for the study of divisibility in various classical rings like GCD domains (including UFD??s), rings of low dimension (including semi-hereditary rings), as well as certain subdirect products of such rings and certain factors of such subdirect products. A Bezout monoid is a commutative monoid S with 0 such that under the natural partial order (for a, b ?? S, a ?? b ?? S ? bS ? aS), S is a distributive lattice, multiplication is distributive over both meets and joins, and for any x, y ?? S, if d = x ?? y and dx ₁ = x then there is a y ₁ ?? S with dy ₁ = y and x ₁ ?? y ₁ = 1. We investigate Bezout monoids by using filters and m-prime filters, and describe all homorphisms between them. We also prove analogues of the Pierce and the Grothendieck sheaf representations of rings for Bezout monoids. The question whether Bezout monoids describe divisibility in Bezout rings (rings whose finitely generated ideals are principal) is still open.     

Conditional limit theorems for critical continuous-state branching processes

We study the conditional limit theorems for critical continuous-state branching processes with branching mechanism ψ(λ) = λ1+αL(1/λ), where α∈ [0, 1] and L is slowly varying at ∞. We prove that if α∈(0, 1], there are norming constants Qt→ 0(as t ↑ +∞) such that for every x 0, Px(QtXt∈·| Xt 0)converges weakly to a non-degenerate limit. The converse assertion is also true provided the regularity of ψ at0. We give a conditional limit theorem for the case α = 0. The limit theorems we obtain in this paper allow infinite variance of the branching process.