On the ranks of the semigroup of A-decreasing finite transformations

2018 ◽  
Vol 17 (02) ◽  
pp. 1850036
Author(s):  
Ping Zhao ◽  
Taijie You ◽  
Huabi Hu
Keyword(s):  
Nonempty Subset ◽  
Idempotent Rank

For [Formula: see text], let [Formula: see text] be the semigroup of all singular mappings on [Formula: see text]. For each nonempty subset [Formula: see text] of [Formula: see text], let [Formula: see text] be the semigroup of all [Formula: see text]-decreasing mappings on [Formula: see text]. In this paper we determine the rank and idempotent rank of the semigroup [Formula: see text].

On the Semigroups of Order-preserving and A-Decreasing Finite Transformations

2014 ◽  
Vol 21 (04) ◽  
pp. 653-662 ◽  
Author(s):  
Ping Zhao
Keyword(s):  
Nonempty Subset ◽  
Abundant Semigroup ◽  
Idempotent Rank

For n ∈ ℕ, let On be the semigroup of all singular order-preserving mappings on [n]= {1,2,…,n}. For each nonempty subset A of [n], let On(A) = {α ∈ On: (∀ k ∈A) kα ≤ k} be the semigroup of all order-preserving and A-decreasing mappings on [n]. In this paper it is shown that On(A) is an abundant semigroup with n-1𝒟*-classes. Moreover, On(A) is idempotent-generated and its idempotent rank is 2n-2- |A\ {n}|. Further, it is shown that the rank of On(A) is equal to n-1 if 1 ∈ A, and it is equal to n otherwise.

On the semigroups of order-preserving transformations generated by idempotents of rank n −1

2017 ◽  
Vol 16 (02) ◽  
pp. 1750023
Author(s):  
Ping Zhao
Keyword(s):  
Nonempty Subset ◽  
Idempotent Rank ◽  
Finite Set

Let [Formula: see text] be the semigroup of all singular order-preserving mappings on the finite set [Formula: see text]. It is known that [Formula: see text] is generated by its set of idempotents of rank [Formula: see text], and its rank and idempotent rank are [Formula: see text] and [Formula: see text], respectively. In this paper, we study the structure of the semigroup generated by any nonempty subset of idempotents of rank [Formula: see text] in [Formula: see text]. We also calculate its rank and idempotent rank.

scholarly journals Hyperfinite and standard unifications for physical theories

2001 ◽  
Vol 28 (2) ◽  
pp. 93-102 ◽  
Author(s):  
Robert A. Herrmann
Keyword(s):  
Nonempty Subset ◽  
Natural Systems ◽  
Consequence Operators

A set of physical theories is represented by a nonempty subset{SNjV|j∈ℕ}of the lattice of consequence operators defined on a languageΛ. It is established that there exists a unifying injection𝒮defined on the nonempty set of significant representations for natural systemsM⊂Λ. IfW∈M, then𝒮Wis a hyperfinite ultralogic and⋃{SNjV(W)|j∈ℕ}=𝒮W(*W)∩Λ. A “product” hyperfinite ultralogicΠis defined on internal subsets of the product set*Λmand is shown to represent the application of𝒮to{W1,…,Wm}⊂M. There also exists a standard unifying injectionSWsuch that𝒮W(*W)⊂*SW(*W).

scholarly journals Decomposable Convexities in Graphs and Hypergraphs

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Francesco M. Malvestuto
Keyword(s):  
Convex Hull ◽  
Convex Sets ◽  
Nonempty Subset ◽  
Convex Set ◽  
Convex Geometry ◽  
Convexity Space ◽  
Vertex Set ◽  
Convex Geometries ◽  
Acyclic Hypergraph ◽  
Maximal Clusters

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).

scholarly journals Green’s Relations on a Semigroup of Transformations with Restricted Range that Preserves an Equivalence Relation and a Cross-Section

Mathematics ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 134
Author(s):  
Chollawat Pookpienlert ◽  
Preeyanuch Honyam ◽  
Jintana Sanwong
Keyword(s):  
Cross Section ◽  
Equivalence Relation ◽  
Nonempty Subset ◽  
Identity Relation ◽  
Green’S Relations ◽  
Restricted Range ◽  
Green's Relations ◽  
Semigroup Of Transformations

Let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed nonempty subset Y of X. For an equivalence relation ρ on X, let ρ^ be the restriction of ρ on Y, R a cross-section of Y/ρ^ and define T(X,Y,ρ,R) to be the set of all total transformations α from X into Y such that α preserves both ρ (if (a,b)∈ρ, then (aα,bα)∈ρ) and R (if r∈R, then rα∈R). T(X,Y,ρ,R) is then a subsemigroup of T(X,Y). In this paper, we give descriptions of Green’s relations on T(X,Y,ρ,R), and these results extend the results on T(X,Y) and T(X,ρ,R) when taking ρ to be the identity relation and Y=X, respectively.

Uniform Harmonic Approximation with Continuous Extension to the Boundary

1988 ◽  
Vol 40 (6) ◽  
pp. 1375-1388 ◽  
Author(s):  
M. Goldstein ◽  
W. H. Ow
Keyword(s):  
Complex Plane ◽  
Nonempty Subset ◽  
Harmonic Approximation ◽  
Continuous Extension ◽  
Partial Derivatives ◽  
Extended Plane ◽  
The One ◽  
Image Position ◽  
One Point Compactification

Let G be a domain in the complex plane and F a nonempty subset of G such that F is the closure in G of its interior F0. We will say f ∊ C1(F) if f is continuous on F and possesses continuous first partial derivatives in F which extend continuously to F as finite-valued functions. Let G* – F be connected and locally connected, f ∊ C1(F) be harmonic in F0, and E be a subset of ∂F ∩ ∂G (here G* denotes the one-point compactification of G and the boundaries ∂F, ∂G are taken in the extended plane). Suppose there is a sequence 〈hn〉 of functions harmonic in G such thatuniformly on F as n → ∞.

On Asymptotic Centers and Fixed Points of Nonexpansive Mappings

1980 ◽  
Vol 32 (2) ◽  
pp. 421-430 ◽  
Author(s):  
Teck-Cheong Lim
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Nonempty Subset ◽  
Nonexpansive Mappings ◽  
Chebyshev Center ◽  
Bounded Subset ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Asymptotic Centers ◽  
Image Position

Let X be a Banach space and B a bounded subset of X. For each x ∈ X, define R(x) = sup{‖x – y‖ : y ∈ B}. If C is a nonempty subset of X, we call the number R = inƒ{R(x) : x ∈ C} the Chebyshev radius of B in C and the set the Chebyshev center of B in C. It is well known that if C is weakly compact and convex, then and if, in addition, X is uniformly convex, then the Chebyshev center is unique; see e.g., [9].Let {Bα : α ∈ ∧} be a decreasing net of bounded subsets of X. For each x ∈ X and each α ∈ ∧, define

A Note on "Implicit Mann Fixed Point Iterations for Pseudo-Contractive Mappings"

2011 ◽  
Vol 393-395 ◽  
pp. 543-545
Author(s):  
Hong Jun Li ◽  
Yong Fu Su
Keyword(s):  
Fixed Point ◽  
Convergence Theorem ◽  
Nonempty Subset ◽  
Applied Mathematics ◽  
Iteration Process ◽  
Implicit Iteration Process ◽  
Contractive Mappings ◽  
Contractive Map ◽  
Fixed Point Iterations ◽  
Implicit Iteration

Ljubomir Ciric, Arif Rafiq, Nenad Cakic, Jeong Sheok Umed [ Implicit Mann fixed point iterations for pseudo-contractive mappings, Applied Mathematics Letters 22 (2009) 581-584] introduced and investigated a modified Mann implicit iteration process for continuous hemi-contractive map. They proved the relatively convergence theorem. However, the content of mann theorem is fuzzy. In this paper, we will give some comments . Let be a Banach space and be a nonempty subset of . A mapping is called hemi-contractive (see [1]) if and In [1], the authors introduced and investigated a modified Mann implicit iteration process for continuous hemi-contractive map. They proved the following convergence theorem.

THE REGULAR PART OF A SEMIGROUP OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE

2017 ◽  
Vol 103 (3) ◽  
pp. 402-419 ◽  
Author(s):  
WORACHEAD SOMMANEE ◽  
KRITSADA SANGKHANAN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Restricted Range ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional ◽  
Idempotent Rank ◽  
Image Position

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.

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