Automorphisms of the endomorphism semigroup of a free commutative dimonoid

We determine all isomorphisms between the endomorphism semigroups of free commutative dimonoids and prove that all automorphisms of the endomorphism semigroup of a free commutative dimonoid are quasi-inner. In particular, we answer a question of B. I. Plotkin.     

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     

A question of B. Plotkin about the semigroup of endomorphisms of a free group

Let be a free group of finite rank , let be the semigroup of endomorphisms of , and let be the group of automorphisms of .

Theorem. If is an automorphism of , then there is an such that for all .

     

On the number of graphs with a given endomorphism monoid

For a given finite monoid , let be the number of graphs on n vertices with endomorphism monoid isomorphic to . For any nontrivial monoid we prove that where and are constants depending only on with .For every k there exists a monoid of size k with , on the other hand if a group of unity of has a size k>2 then .     

Split graphs whose half-strong endomorphisms form a monoid

In this paper,the half-strong,the locally strong and the quasi-strong endomorphisms of a split graph are investigated.Let X be a split graph and let End(X),hEnd(X),lEnd(X) and qEnd(X) be the endomorphism monoid,the set of all half-strong endomorphisms,the set of all locally strong endomorphisms and the set of all quasi-strong endomorphisms of X,respectively.The conditions under which hEnd(X) forms a submonoid of End(X) are given.It is shown that lEnd(X) = qEnd(X) for any split graph X.The conditions under which lEnd(X)(resp.qEnd(X)) forms a submonoid of End(X) are also given.In particular,if hEnd(X) forms a monoid,then lEnd(X)(resp.qEnd(X)) forms a monoid too.     

Endomorphisms of automorphism groups of free groups

It is proved that any non-trivial endomorphism of an automorphism group AutF_n of a free group F_n, for n 3, either is an automorphism or factorization over a proper automorphism subgroup. An endomorphism of AutF₂ is an automorphism, or else a homomorphism onto one of the groups S₃, D₈, Z₂ × Z₂, Z₂, or (Z₂ × Z₂). A non-trivial homomorphism of AutF_n into AutF_m, for n 3, m 2, and n > m, is a homomorphism onto Z₂ with kernel SAutF_n. As a consequence, we obtain that AutF_n is co-Hopfian.Supported by RFBR grant No. 02-01-00293 and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 211–237, March–April, 2005.     

Absolute graphs with prescribed endomorphism monoid

We consider endomorphism monoids of graphs. It is well-known that any monoid can be represented as the endomorphism monoid M of some graph Γ with countably many colors. We give a new proof of this theorem such that the isomorphism between the endomorphism monoid $\mathop{\rm End}\nolimits (\Gamma)We consider endomorphism monoids of graphs. It is well-known that any monoid can be represented as the endomorphism monoid M of some graph Γ with countably many colors. We give a new proof of this theorem such that the isomorphism between the endomorphism monoid and M is absolute, i.e. holds in any generic extension of the given universe of set theory. This is true if and only if |M|,|Γ| are smaller than the first Erdős cardinal (which is known to be strongly inaccessible). We will encode Shelah’s absolutely rigid family of trees (Isr. J. Math. 42(3), 177–226, 1982) into Γ. The main result will be used to construct fields with prescribed absolute endomorphism monoids, see G?bel and Pokutta (Shelah’s absolutely rigid trees and absolutely rigid fields, in preparation). This work was supported by the project No. I-706-54.6/2001 of the German-Israeli Foundation for Scientific Research & Development and a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD).     

Automorphisms of endomorphism semigroups of reflexive digraphs

A reflexive digraph is a pair (X, ρ), where X is an arbitrary set and ρ is a reflexive binary relation on X. Let End (X, ρ) be the semigroup of endomorphisms of (X, ρ). We determine the group of automorphisms of End (X, ρ) for: digraphs containing an edge not contained in a cycle, digraphs consisting of arbitrary unions of cycles such that cycles of length ≥2 are pairwise disjoint, and some circulant digraphs (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The join of split graphs whose half-strong endomorphisms form a monoid

In this paper, the half-strong endomorphisms of the join of split graphs are investigated. We give the conditions under which the half-strong endomorphisms of the join of split graphs form a monoid.     

Construction of lattice orders on the semigroup ring of a positive rooted monoid

Lattice orders on the semigroup ring of a positive rooted monoid are constructed, and it is shown how to make the monoid ring into a lattice-ordered ring with squares positive in various ways. It is proved that under certain conditions these are all of the lattice orders that make the monoid ring into a lattice-ordered ring. In particular, all of the partial orders on the polynomial ring A[x] in one positive variable are determined for which the ring is not totally ordered but is a lattice-ordered ring with the property that the square of every element is positive. In the last section some basic properties of d-elements are considered, and they are used to characterize lattice-ordered division rings that are quadratic extensions of totally ordered division rings.     

The endomorphism monoid of a free trioid of rank 1

We describe all endomorphisms of a free trioid of rank 1 and construct a semigroup which is isomorphic to the endomorphism monoid of such free trioid. Also, we give an abstract characteristic for the endomorphism monoid of a free trioid of rank 1 and prove that free trioids are determined by their endomorphism monoids.     

交换环上严格上三角矩阵半群的自同构(英文)

设R是任意含有单位元的交换环,N_n(R)是R上的所有n×n严格上三角矩阵组成的乘法半群.本文对N_n(R)的任意自同构给出了详细的描述.     

半环上的三角矩阵乘法半群的自同构

设R是含有恒等元1的半环,C是R上的中心子半环.Tn(R)是R上的n阶上三角矩阵C-代数.证明了当R是一个幂等元都是中心元的半环时,映射Φ:Tn(R)→Tn(R)是乘法半群自同构当且仅当存在Tn(R)中的可逆矩阵G和R中的半环自同构τ使得A=(aij)n×n∈Tn(R),均有Φ(A)=G-1τ(A)G.这里τ(A)=(τ(aij))n×n,n2.     

On the topological entropy of the free semigroup action generated by proper maps

In this paper, we introduce the topological entropy of a free semigroup action generated by proper maps, which extends the notions of the topological entropy of the free semigroup actions defined by Bufetov in 1999 and topological entropy of the proper maps defined by Patrão in 2010. We then give some properties of these notions and discuss the relations between them. We also give a partial variational principle for locally compact separable metric spaces. Moreover, the relationship between topological entropy of the free semigroup generated by proper maps and topological entropy of a skew-product transformation is given. These results extend the results obtained by Patrão, Bufetov and Lin, Ma and Wang in 2018.