scholarly journals Certain homomorphisms of a compact semigroup onto a thread

1967 ◽  
Vol 7 (3) ◽  
pp. 311-322 ◽  
Author(s):  
R. P. Hunter ◽  
L. W. Anderson
Keyword(s):  
Nilpotent Element ◽  
Identity Element ◽  
Compact Semigroup ◽  
Continuous Homomorphism ◽  
Zero Element ◽  
Special Role ◽  
Special Aspect ◽  
Rees Quotient ◽  
Usual Product ◽  
The Ideal

Let S be a compact semigroup and f a continuous homomorphism of S onto the (compact) semigroup T. What can be said concerning the relations among S, f, and T? It is to one special aspect of this problem which we shall address ourselves. In particular, our primary considerations will be directed toward the case in which T is a standard thread. A standard thread is a compact semigroup which is topologically an arc, one endpoint being an identity element, the other being a zero element. The structure of standard threads is rather completely determined e.g. see [20]. Among the standard threads there are three which have a rather special rôle. These are as follows: A unit thread is a standard thread with only two idempotents and no nilpotent element. A unit thread is isomorphic to the usual unit interval [14]. A nil thread again has only two idempotents but has a non-zero nilpotent element. A nil thread is isomorphic with the interval [½, 1], the multiplication being the maximum of ½ and the usual product — or, what is the same thing, the Rees quotient of the usual [0, 1] by the ideal [0,½ ]. Finally there is the idempotent thread, the multiplication being x o y = mm (x, y). These three standard threads can often be considered separately and, in this paper, we reserve the symbols I1I2 and I3 to denote the unit, nil and idempotent threads respectively. Also, throughout this paper, by a homomorphism we mean a continuous homomorphism.

FINITELY BASED WORDS

2000 ◽  
Vol 10 (04) ◽  
pp. 457-480 ◽  
Author(s):  
OLGA SAPIR
Keyword(s):  
Sufficient Conditions ◽  
Basis Property ◽  
Free Monoid ◽  
Finite Basis ◽  
Finitely Based ◽  
Finite Basis Property ◽  
Letter Alphabet ◽  
Finite Language ◽  
Rees Quotient ◽  
The Ideal

Let W be a finite language and let Wc be the closure of W under taking subwords. Let S(W) denote the Rees quotient of a free monoid over the ideal consisting of all words that are not in Wc. We call W finitely based if the monoid S(W) is finitely based. Although these semigroups have easy structure they behave "generically" with respect to the finite basis property [6]. In this paper, we describe all finitely based words in a two-letter alphabet. We also find some necessary and some sufficient conditions for a set of words to be finitely based.

scholarly journals Near-rings of continuous functions on disconnected groups

1979 ◽  
Vol 28 (4) ◽  
pp. 433-451 ◽  
Author(s):  
Robert D. Hofer
Keyword(s):  
Topological Group ◽  
Identity Element ◽  
Continuous Functions ◽  
Ideal Structure ◽  
Rings Of Continuous Functions ◽  
The Ideal

AbstractN(G) denotes the near-ring of all continuous selfmaps of the topological group G (under composition and the pointwise induced operation) and N0(G) is the subnear-ring of N(G) consisting of all functions having the identity element of G fixed. It is known that if G is discrete then (a) N0(G) is simple and (b) N(G) is simple if and only if G is not of order 2. We begin a study of the ideal structure of these near-rings when G is a disconnected group.

Degree Functions in Local Rings

1961 ◽  
Vol 57 (1) ◽  
pp. 1-7 ◽  
Author(s):  
D. Rees
Keyword(s):  
Maximal Ideal ◽  
Zero Element ◽  
Transcendence Degree ◽  
Local Rings ◽  
Local Domain ◽  
Degree Function ◽  
Finite Set ◽  
Residue Fields ◽  
Image Position ◽  
The Ideal

Let Q be a local domain of dimension d with maximal ideal m and let q be an m-primary ideal. Then we define the degree function dq(x) to be the multiplicity of the ideal , where x; is a non-zero element of m. The degree function was introduced by Samuel (5) in the case where q = m. The function dq(x) satisfies the simple identityThe main purpose of this paper is to obtain a formulawhere vi(x) denotes a discrete valuation centred on m (i.e. vi(x) ≥ 0 if x ∈ Q, vi(x) > 0 if x ∈ m) of the field of fractions K of Q. The valuations vi(x) are assumed to have the further property that their residue fields Ki have transcendence degree d − 1 over k = Q/m. The symbol di(q) denotes a non-negative integer associated with vi(x) and q which for fixed q is zero for all save a finite set of valuations vi(x).

The Group Ring Of a Class Of Infinite Nilpotent Groups

1955 ◽  
Vol 7 ◽  
pp. 169-187 ◽  
Author(s):  
S. A. Jennings
Keyword(s):  
Nilpotent Group ◽  
Group Ring ◽  
Characteristic Zero ◽  
Discrete Group ◽  
Nilpotent Groups ◽  
Zero Element ◽  
Free Nilpotent Group ◽  
Image Position ◽  
Torsion Free ◽  
The Ideal

Introduction. In this paper we study the (discrete) group ring Γ of a finitely generated torsion free nilpotent group over a field of characteristic zero. We show that if Δ is the ideal of Γ spanned by all elements of the form G − 1, where G ∈ , thenand the only element belonging to Δw for all w is the zero element (cf. (4.3) below).

FINITE HIGHER COMMUTATORS IN ASSOCIATIVE RINGS

2013 ◽  
Vol 89 (3) ◽  
pp. 503-509
Author(s):  
CHARLES LANSKI
Keyword(s):  
Nilpotent Element ◽  
Associative Ring ◽  
Minimal Cardinality ◽  
Associative Rings ◽  
The Ideal

AbstractIf $T$ is any finite higher commutator in an associative ring $R$, for example, $T= [[R, R] , [R, R] ] $, and if $T$ has minimal cardinality so that the ideal generated by $T$ is infinite, then $T$ is in the centre of $R$ and ${T}^{2} = 0$. Also, if $T$ is any finite, higher commutator containing no nonzero nilpotent element then $T$ generates a finite ideal.

Quasi-automatic semigroups

2020 ◽  
pp. 2150046
Author(s):  
Yanhui Wang ◽  
Yuhan Wang ◽  
Xueming Ren ◽  
Kar Ping Shum
Keyword(s):  
Cayley Graph ◽  
Identity Element ◽  
Zero Element ◽  
The Other ◽  
Finitely Generated ◽  
Converse Statement ◽  
Free Products ◽  
Generating Set ◽  
Automatic Semigroup ◽  
Automatic Group

Quasi-automatic semigroups are extensions of a Cayley graph of an automatic group. Of course, a quasi-automatic semigroup generalizes an automatic semigroup. We observe that a semigroup [Formula: see text] may be automatic only when [Formula: see text] is finitely generated, while a semigroup may be quasi-automatic but it is not necessary finitely generated. Similar to the usual automatic semigroups, a quasi-automatic semigroup is closed under direct and free products. Furthermore, a semigroup [Formula: see text] is graph automatic if and only if [Formula: see text] with a zero element adjoined is graph automatic, and also a semigroup [Formula: see text] is graph automatic if and only if [Formula: see text] with an identity element adjoined is graph automatic. However, the class of quasi-automatic semigroups is a much wider class than the class of automatic semigroups. In this paper, we show that every automatic semigroup is quasi-automatic but the converse statement is not true (see Example 3.6). In addition, we notice that the quasi-automatic semigroups are invariant under the changing of generators, while a semigroup may be automatic with respect to a finite generating set but not the other. Finally, the connection between the quasi-automaticity of two semigroups [Formula: see text] and [Formula: see text], where [Formula: see text] is a subsemigroup with finite Rees index in [Formula: see text] will be investigated and considered.

A note on the generalized nilpotent elements of a module

2020 ◽  
pp. 289-294
Author(s):  
Saugata Purkayastha ◽  
Helen K. Saikia
Keyword(s):  
Nilpotent Element ◽  
Zero Element ◽  
Nilpotent Elements ◽  
Strongly Nilpotent

In this paper, we introduce the notion of the generalized nilpotent element of a module. In \cite{Groenewald}, the notion of nilpotent element of a module is introduced in the following sense: a non-zero element $m$ of an $R$-module $M$ is said to be nilpotent if there exists some $a\in R$ such that $a^k m=0$ but $am\neq 0$ for some $k\in \mathbb N$. In our present work we aim to generalize this notion. We have extended this notion to the strongly nilpotent element of a module.

scholarly journals Inverse semigroups generated by nilpotent transformations

1984 ◽  
Vol 99 (1-2) ◽  
pp. 153-162 ◽  
Author(s):  
John M. Howie ◽  
M. Paula O. Marques-Smith
Keyword(s):  
Inverse Semigroup ◽  
Homomorphic Image ◽  
Inverse Semigroups ◽  
Nilpotent Elements ◽  
Inverse Subsemigroup ◽  
Symmetric Inverse Semigroup ◽  
Infinite Cardinality ◽  
Index 2 ◽  
Rees Quotient ◽  
The Ideal

SynopsisLet X be a set with infinite cardinality m and let B be the Baer-Levi semigroup, consisting of all one-one mappings a:X→X for which ∣X/Xα∣ = m. Let Km=<B 1B>, the inverse subsemigroup of the symmetric inverse semigroup ℐ(X) generated by all products β−γ, with β,γ∈B. Then Km = <N2>, where N2 is the subset of ℐ(X) consisting of all nilpotent elements of index 2. Moreover, Km has 2-nilpotent-depth 3, in the sense that Let Pm be the ideal {α∈Km: ∣dom α∣<m} in Km and let Lm be the Rees quotient Km/Pm. Then Lm is a 0-bisimple, 2-nilpotent-generated inverse semigroup with 2-nilpotent-depth 3. The minimum non-trivial homomorphic image of Lm also has these properties and is congruence-free.

Jean Delville's La Mission de l'Art: Hegelian Echoes in fin-de-siècle Idealism

2007 ◽  
Vol 11 (3-4) ◽  
pp. 330-372
Author(s):  
Brendan Cole
Keyword(s):  
Nineteenth Century ◽  
Natural World ◽  
Late Nineteenth Century ◽  
Arthur Schopenhauer ◽  
Special Role ◽  
Prolific Author ◽  
Physical Form ◽  
The Subject ◽  
Main Ideas ◽  
The Ideal

AbstractJean Delville was not only a gifted painter, but also a prolific author, poet and polemicist. He is unique amongst his artistic contemporaries for having written extensively on the subject of Idealism in art. Idealist philosophy, as an intellectual influence, was fairly pervasive amongst contemporary non-realist authors, poets and painters; the core nineteenth-century influence in this regard was the writings of Arthur Schopenhauer. Delville, however, took a different path, particularly in his seminal book, La Mission de l'Art, and his various polemical essays on the subject, which reflect, rather, key ideas derived from the writings of the German Idealist, G.W.F. Hegel. Hegel's influence on late-nineteenth century non-realist art is understated in the literature. This paper analyses the main ideas of Delville's La Mission de l'Art in the context of Hegelian Idealism. It focuses on key areas of this tradition, specifically with regard to the nature of the Idea and the Ideal, the relation of the Ideal to the natural world, the relation between the Idea and the notion of Beauty and the special role of the artist in revealing the Idea in physical form.

FINITELY BASED, FINITE SETS OF WORDS

2000 ◽  
Vol 10 (06) ◽  
pp. 683-708 ◽  
Author(s):  
MARCEL JACKSON ◽  
OLGA SAPIR
Keyword(s):  
Direct Product ◽  
Free Monoid ◽  
Finitely Based ◽  
Finite Sets ◽  
Finite Semigroups ◽  
Finite Set ◽  
Rees Quotient ◽  
The Ideal

For W a finite set of words, we consider the Rees quotient of a free monoid with respect to the ideal consisting of all words that are not subwords of W. This resulting monoid is denoted by S(W). It is shown that for every finite set of words W, there are sets of words U⊃W and V⊃W such that the identities satisfied by S(V) are finitely based and those of S(U) are not finitely based [regardless of the situation for S(W)]. The first examples of finitely based (not finitely based) aperiodic finite semigroups whose direct product is not finitely based (finitely based) are presented and it is shown that every monoid of the form S(W) with fewer than 9 elements is finitely based and that there is precisely one not finitely based 9 element example.

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