On the injective envelope of a graded module

Abstract

Let Sing _n be the semigroup of all singular full transformations on the set X _n  = {1, 2,…, n} under the composition of functions. Let E(J _n ? 1) be the set of all idempotents of the top 𝒥-class J _n ? 1 = {α ∈ Sing _n :|im α| = n-1}. For any nonempty subset I of E(J _n  ? 1), the aim of this paper is to find a constructive necessary and sufficient condition for the semiband S(I) = ?I? to be ?-trivial. Further, the semiband S(I) is locally maximal ?-trivial if S(I) is ?-trivial and S(I ∪ {e}) is not ?-trivial for any e ∈ E(J _n ? 1 )\I. As applications, we classify locally maximal ?-trivial subsemibands and locally maximal regular ?-trivial subsemibands of Sing _n , respectively. Moreover, the characterization of which S(I) is a band is obtained.

     

Maximal regular subsemibands of SOP n

We describe the maximal regular subsemibands of the finite singular orientation-preserving transformation semigroup SOP _n and completely obtain their classification.     

A classification of the maximal idempotent-generated subsemigroups of finite singular semigroups

We describe the maximal idempotent-generated subsemigroups of the finite singular semigroup Sing _n on the finite set X _n ={1,2, \ldots,n} , and count the number of its maximal idempotent-generated subsemigroups. October 21, 1999     

Maximal regular subsemibands of finite order-preserving transformation semigroups <Emphasis Type="Italic">K</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">r</Emphasis>)

Let O _n be the order-preserving transformation semigroup on X _n . For an arbitrary integer r such that 1≤r≤n−2, we completely describe the maximal regular subsemibands of the semigroup K(n,r)={α∈O _n :|im(α)|≤r}. We also formulate the cardinal number of such subsemigroups.     

Fully invariant transformations and associated groups

It is well known that the symmetric group S _ntogether with one idempotent of rank n- 1 on a finite n-element set Nserves as a set of generators for the semigroup T _nof all the total transformations on N. It is also well known that the singular part Sing _n of T _n can be generated by a set of idempotents of rank n- 1. The purpose of this paper is to begin an investigation of the way in which Sing_nand its subsemigroups can be generated by the conjugates of a subset of elements of T _n by a subgroup of S _n . We look for the smallest subset of elements of T _n that will serve and, correspondingly, for a characterization of those subgroups of S _n that will serve. Using some techniques from graph theory we prove our main result:the conjugates of a single transformation of rank n- 1 under Gsuffice to generate Sing_nif and only if Gis what we define to be a 2-block transitive subgroup of S _n .     

Maximal properties of some subsemigroups in finite order-preserving transformation semigroups

Let [n] = {1,2,…,n} be a finite set, ordered in the usual way. The order-preserving transformation semigroup O_n is the set of all order-preserving transformations of [n] (excluding the identity mapping) under composition. In this paper we first describe maximal idempotent-generated subsemigroups of O _n, and show that O_n has 2n - 2 such subsemi-groups. Secondly, we investigate maximal regular subsemigroups of O_n , and obtain the number of such subsemigroups as 2n - 3. Thirdly, we describe maximal idempotent-generated regular subsemigroups of O_n , and also obtain their classification and number.     

On the Monoid of All Partial Order-Preserving Extensive Transformations

A partial transformation α on an n-element chain X _n is called order-preserving if x ≤ y implies xα ≤yα for all x, y in the domain of α and it is called extensive if x ≤ xα for all x in the domain of α. The set of all partial order-preserving extensive transformations on X _n forms a semiband POE _n . We determine the maximal subsemigroups as well as the maximal subsemibands of POE _n .     

Isodiametric Problems for Polygons

The maximal area of a polygon with n = 2m edges and unit diameter is not known when m ≥ 5, nor is the maximal perimeter of a convex polygon with n = 2^m edges and unit diameter known when m ≥ 4. We construct improved polygons in both problems, and show that the values we obtain cannot be improved for large n by more than c₁/n³ in the area problem and c₂/n⁵ in the perimeter problem, for certain constants c₁ and c₂.     

Spanning maximal planar subgraphs of random graphs

We study the threshold for the existence of a spanning maximal planar subgraph in the random graph G_{n, p} . We show that it is very near p = 1/n^? We also discuss the threshold for the existence of a spanning maximal outerplanar subgraph. This is very near p = 1/n^½.     

Structure of the semigroup <Emphasis Type="Italic">OT</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We study the structure of the semigroup OT _n , which is a unique (up to an isomorphism) R-section of the semigroup T _n . For this semigroup, we describe Green relations, determine regular and nilpotent elements, describe maximal nilpotent subsemigroups, and determine the unique irreducible system of generatrices and maximal subsemigroups.     

On maximal triangle-free graphs

A triangle-free graph is maximal if adding any edge will create a triangle. The minimal number of edges of a maximal triangle-free graph on n vertices having maximal degree at most D is denoted by F(n, D). We determine the value of lim_n-∞ F(n, cn)/n for 2/5 < c < 1/2. This investigation continues work done by Z. Füredi and Á. Seress. Our result is contrary to a conjecture of theirs.     

A note on maximal regular subsemigroups of the finite transformation semigroups \mathcal{T}(n,r)

Let $\mathcal{T}_{n}$ be the semigroup of all full transformations on the finite set X _n ={1,2,…,n}. For 1≤r≤n, set $\mathcal {T}(n, r)=\{ \alpha\in\mathcal{T}_{n} | \operatorname{rank}(\alpha)\leq r\}$ . In this note we show that, for 2≤r≤n?2, any maximal regular subsemigroup of the semigroup $\mathcal{T} (n,r)$ is idempotent generated, but this may not happen in the semigroup $\mathcal{T}(n, n-1)$ .     

Rank Properties of the Semigroup of Singular Transformations on a Finite Set

It is known that the semigroup Sing _n of all singular self-maps of X _n  = {1,2,…, n} has rank n(n ? 1)/2. The idempotent rank, defined as the smallest number of idempotents generating Sing _n , has the same value as the rank. (See Gomes and Howie, 1987 Gomes , G. M. S. , Howie , J. M. ( 1987 ). On the rank of certain finite semigroups of transformations . Math. Proc. Cambridge Phil. Soc. 101 : 395 – 303 .[Crossref], [Web of Science ®] , [Google Scholar].) Idempotents generating Sing _n can be seen as special cases (with m = r = 2) of (m, r)-path-cycles, as defined in Ay\i k et al. (2005 Ay?k , G. , Ay?k , H. , Howie , J. M. ( 2005 ). On factorisations and generators in transformation semigroups . Semigroup Forum 70 : 225 – 237 .[Crossref], [Web of Science ®] , [Google Scholar]). The object of this article is to show that, for fixed m and r, the (m, r)-rank of Sing _n , defined as the smallest number of (m, r)-path-cycles generating Sing _n , is once again n(n ? 1)/2.     

Minimal regular 2-graphs and applications

A 2-graph is a hypergraph with edge sizes of at most two. A regular 2-graph is said to be minimal if it does not contain a proper regular factor. Let f2(n) be the maximum value of degrees over all minimal regular 2-graphs of n vertices. In this paper, we provide a structure property of minimal regular 2-graphs, and consequently, prove that f2(n) = n 3-i/3, where 1 ≤ i ≤ 6, i ≡ n (mod 6) and n ≥ 7, which solves a conjecture posed by Fan, Liu, Wu and Wong. As applications in graph theory, we are able to characterize unfactorable regular graphs and provide the best possible factor existence theorem on degree conditions. Moreover, fa(n) and the minimal 2-graphs can be used in the universal switch box designs, which originally motivated this study.     

ON A CONJECTURE ON nTH ORDER DEGREE REGULAR GRAPHS

Abstract

For n a positive integer and v a vertex of a graph G, the nth order degree of v in G, denoted by deg_nv, is the number of vertices at distance n from v. The graph G is said to be nth order regular of degree k if, for every vertex v of G, deg_nv = k. The following conjecture due to Alavi, Lick, and Zou is proved: For n ≥ 2, if G is a connected nth order regular graph of degree 1, then G is either a path of length 2n—1 or G has diameter n. Properties of nth order regular graphs of degree k, k ≥ 1, are investigated.     

Spaces with a Souslin and a shanin condition

If X is a regular hereditary Souslin space and x ∈X then either there exists a sequence {x_n: n=1, 2, ...} ? X{x} such that x ∈ [{x_n∶n=1, 2, ...}], or the pseudocharacter of x in X is no greater than countable. In other words, if X is a hereditary Souslin bicompactum which is a χ-space, then X is a Frechet-Urysohn space.     

Fine and Wilf words for any periods

Let w = w₁ … w_n be a word of maximal length n, and with a maximal number of distinct letters for this length, such that w has periods p₁, …, p_n but not period gcd(p₁,…,p_r). We provide a fast algorithm to compute n and w. We show that w is uniquely determined apart from isomorphism and that it is a palindrome. Furthermore we give lower and upper bounds for n as explicit functions of p₁, …p_r. For r = 2 the exact value of n is due to Fine and Wilf. In case the number of distinct letters in the extremal word equals r a formula for n had been given by Castelli, Mignosi and Restivo in case r = 3 and by Justin if r > 3.     

A family of Bernstein quasi-interpolants on [0,1]

Suppose that we want to approximate f∈C[0,1] by polynomials inP, using only its values on X_n={i/n, 0≤i≤n}. This can be done by the Lagrange interpolant L_n f or the classical Bernstein polynomial B_n f. But, when n tends to infinity, L_n f does not converge to f in general and the convergence of B_n f to f is very slow. We define a family of operators B _n ^(k) , n≥k, which are intermediate ones between B _n ⁽⁰⁾ =B _n ⁽¹⁾ =B_n and B _n ⁽ⁿ⁾ =L_n, and we study some of their properties. In particular, we prove a Voronovskaja-type theorem which asserts that B _n ^(k) f−f=O(n^−[(k+2)/2]) for f sufficiently regular. Moreover, B _n ^(k) f uses only values of B_n f and its derivaties and can be computed by De Casteljau or subdivision algorithms.     

Integral Domains with Highly Nonunique Factorization

For a positive integer n, an atomic integral domain R is defined to be completely non- n- factorial if for any n atoms a₁…, a_n, the product a₁ … a _n has as highly nonunique a factorization into atoms as possible in that given any n ? 1 atoms b₁,…, b_{nt - 1}, b₁ … b _n? 1¦a₁ … a _n. We show that R is completely non-n-factorial for some n ≥ 2 if and only if (R, M) is a quasilocal domain with [M: M] a DVR having M as its maximal ideal.