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.     

Weakly Integrally Closed Numerical Monoids and Forbidden Patterns

An integral domain D is weakly integrally closed if whenever x is in the quotient field of D, and J is a nonzero finitely generated ideal of D such that xJ ? J ², then x is in D. We define weakly integrally closed (WIC) numerical monoids similarly. If a monoid algebra is weakly integrally closed, then so is the monoid. The characteristic function of a numerical monoid M can be thought of as an infinite binary string s(M). A pattern of finitely many 0's and 1's is called forbidden if whenever s(M) contains it, then M is not weakly integrally closed. The pattern 11011 is forbidden. We show that a numerical monoid M is WIC if and only if s(M) contains no forbidden patterns. We also show that for every finite set S of forbidden patterns, there exists a numerical monoid M that is not WIC and for which s(M) contains no stretch (in a natural sense) of a pattern in S.     

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.     

Graph expansions of unipotent monoids

Margolis and Meakin use the Cayley graph of a group presentation to construct E-unitary inverse monoids [11]. This is the technique we refer to as graph expansion. In this paper we consider graph expansions of unipotent monoids, where a monoid is unipotent if it contains a unique idempotent. The monoids arising in this way are E-unitary and belong to the quasivariety of weakly left ample monoids. We give a number of examples of such monoids. We show that the least unipotent congruence on a weakly left ample monoid is given by the same formula as that for the least group congruence on an inverse monoid and we investigate the notion of proper for weakly left ample monoids.

Using graph expansions we construct a functor F^e from the category U of unipotent monoids to the category PWLA of proper weakly left ample monoids. The functor F^e is an expansion in the sense of Birget and Rhodes [2]. If we equip proper weakly left ample monoids with an extra unary operation and denote the corresponding category by PWLA ⁰ then regarded as a functor U→PWLA ⁰ F^e is a left adjoint of the functor F^σ : PWLA ⁰ → U that takes a proper weakly left ample monoid to its greatest unipotent image.

Our main result uses the covering theorem of [8] to construct free weakly left ample monoids.     

Monoids and Groups of <Emphasis Type="Italic">I</Emphasis>-Type

A monoid S generated by {x₁,. . .,x_n} is said to be of (left) I-type if there exists a map v from the free Abelian monoid FaM_n of rank n generated by {u₁,. . .,u_n} to S so that for all a∈FaM_n one has {v(u₁a),. . .,v(u_na)}={x₁v(a),. . .,x_nv(a)}. Then S has a group of fractions, which is called a group of (left) I-type. These monoids first appeared in the work of Gateva-Ivanova and Van den Bergh, inspired by earlier work of Tate and Van den Bergh. In this paper we show that monoids and groups of left I-type can be characterized as natural submonoids and groups of semidirect products of the free Abelian group Fa_n and the symmetric group of degree n. It follows that these notions are left–right symmetric. As a consequence we determine many aspects of the algebraic structure of such monoids and groups. In particular, they can often be decomposed as products of monoids and groups of the same type but on less generators and many such groups are poly-infinite cyclic. We also prove that the minimal prime ideals of a monoid S of I-type, and of the corresponding monoid algebra, are principal and generated by a normal element. Further, via left–right divisibility, we show that all semiprime ideals of S can be described. The latter yields an ideal chain of S with factors that are semigroups of matrix type over cancellative semigroups. In memory of Paul Wauters Mathematics Subject Classifications (2000) 20F05, 20M05; 16S34, 16S36, 20F16. The authors were supported in part by Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium), Flemish–Polish bilateral agreement BIL 01/31, and KBN research grant 2P03A 033 25 (Poland).     

On flatness of acts

We consider monoids $S=G\dot \cup I$ where G is a group and I is an ideal of S and show that if an S-act is principally weakly flat, (weakly) flat, torsion free or satisfies conditions (P) or (P_E) as an I¹-act, then it has these properties as an S-act. We also show that an S-act which is free, projective or strongly flat as an I¹-act may not generally have these properties as an S-act.     

On the Space of Weakly Almost Periodic Functionals and Its Amenability

Let S be a locally compact semitopological semigroup with measure algebra M(S), M₀(S) the set of all probability measures in M(S) and WF(S) the space of weakly almost periodic functionals on M(S)^*. Assuming that M₀(S) has the semiright invariant isometry property, it is shown that WF(S) has a topological left invariant mean (TLIM) whenever the center of M₀(S) is nonempty; in particular if either the center of S is nonempty or S has a left identity, then WF(S) has a TLIM. Finally if, for each M₀(S), the mapping v v * of M₀(S) into itself is surjective and the center of M₀(S) is nonempty, then WF(S) has a TLIM. We also generalize some results from discrete case to topological one.AMS Subject Classification (1991): 43A07     

Strongly Flat and Condition (P) Covers of Acts Over Monoids

In this article we characterize monoids over which every right S-act has a strongly flat (condition (P)) cover. Similar to the perfect monoids, such monoids are characterized by condition (A) and having strongly flat (condition (P)) cover for each cyclic right S-act. We also give a new characterization for perfect monoids as monoids over which every strongly flat right S-act has a projective cover.     

Flatness properties of diagonal acts over monoids

If S is a monoid, the right S-act S×S, equipped with componentwise S-action, is called the diagonal act of S. The question of when this act is cyclic or finitely generated has been a subject of interest for many years, but so far there has been no explicit work devoted to flatness properties of diagonal acts. Considered as a right S-act, the monoid S is free, and thus is also projective, flat, weakly flat, and so on. In 1991, Bulman-Fleming gave conditions on S under which all right acts S ^I (for I a non-empty set) are projective (or, equivalently, when all products of projective right S-acts are projective). At approximately the same time, Victoria Gould solved the corresponding problem for strong flatness. Implicitly, Gould’s result also answers the question for condition (P) and condition (E). For products of flats, weakly flats, etc. to again have the same property, there are some published results as well. The specific questions of when S×S has certain flatness properties have so far not been considered. In this paper, we will address these problems. S. Bulman-Fleming research supported by Natural Sciences and Engineering Research Council of Canada Research Grant A4494. Some of the results in this article are contained in the M.Math. thesis of A. Gilmour, University of Waterloo (2007).     

(Homo) Flatness of posets on poideal extensions

We consider pomonoids , where G is a pogroup and I is a poideal of S and show that if an S-poset is principally weakly flat, (weakly) flat, po-flat, (principally) weakly po-flat, (po-) torsion free or satisfies Conditions (P), (P _E ), (P _w ), (PWP), (PWP) _w , (WP) or (WP) _w as an I ¹-poset, then it has these properties as an S-poset. We also show that an S-poset which is free, projective or strongly flat as an I ¹-poset may not generally have these properties as an S-poset.     

Weakly left localizable rings

A new class of rings, the class of weakly left localizable rings, is introduced. A ring R is called weakly left localizable if each non-nilpotent element of R is invertible in some left localization S^?1R of the ring R. Explicit criteria are given for a ring to be a weakly left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case for all left Noetherian rings). It is proved that a ring with finitely many maximal left denominator sets that satisfies some natural conditions is a weakly left localizable ring iff its left quotient ring is a direct product of finitely many local rings such that their radicals are nil ideals.     

Gorenstein flatness and injectivity over Gorenstein rings

Let R be a Gorenstein ring.We prove that if I is an ideal of R such that R/I is a semi-simple ring,then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical.In addition,we prove that if R→S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules,then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical.We also give some applications of these results.     

Monoids over which all strongly flat left acts are regular

Let Sbe a monoid. It is shown that all strongly flat left S-acts are regular if and only if all left S-acts having the property (E) are regular if and only if Sis a left PP monoid and satisfies (FP₂).This result answers a question in Kilp and Knauer [5].     

On Monoids for Which all Condition (P) Acts are Strongly Flat

In this article, we continue to investigate the monoids over which all right S-acts satisfying condition (P) are strongly flat, and we obtain some new classes of monoids, thereby extending all previous results in this area.     

Weakly Outer Inner Functions

An inner function I in the unit ball B_nB_nⁿ is said to be weakly outer if the closed subspace I H ^p(B ⁿ) is weakly dense in the Hardy space H^p(B ⁿ), 0n for all n1. We also investigate inner functions I such that the subspace IH^p(B ⁿ) is not weakly dense in H^p(B ⁿ).     

On graphs with a given endomorphism monoid

Hedrlín and Pultr proved that for any monoid M there exists a graph G with endomorphism monoid isomorphic to M . In this paper we give a construction G(M) for a graph with prescribed endomorphism monoid M . Using this construction we derive bounds on the minimum number of vertices and edges required to produce a graph with a given endomorphism monoid for various classes of finite monoids. For example we show that for every monoid M , | M |=m there is a graph G with End(G)? M and |E(G)|≤(1 + 0(1))m². This is, up to a factor of 1/2, best possible since there are monoids requiring a graph with \begin{eqnarray*} && \frac{m^{2}}{2}(1 -0(1)) \end{eqnarray*} edges. We state bounds for the class of all monoids as well as for certain subclasses—groups, k‐cancellative monoids, commutative 3‐nilpotent monoids, rectangular groups and completely simple monoids. © 2009 Wiley Periodicals, Inc. J Graph Theory 62, 241–262, 2009