Independence algebras, basis algebras and semigroups of quotients

2010 ◽  
Vol 53 (3) ◽  
pp. 697-729 ◽  
Author(s):  
Victoria Gould
Keyword(s):  
Endomorphism Monoid ◽  
Left Order

AbstractWe show that if A is a stable basis algebra satisfying the distributivity condition, then B is a reduct of an independence algebra A having the same rank. If this rank is finite, then the endomorphism monoid of B is a left order in the endomorphism monoid of A.

scholarly journals Products of two idempotent transformations over arbitrary sets and vector spaces

1998 ◽  
Vol 57 (1) ◽  
pp. 59-71 ◽  
Author(s):  
Rachel Thomas
Keyword(s):  
Vector Space ◽  
Transformation Semigroup ◽  
Vector Spaces ◽  
Endomorphism Monoid ◽  
Linear Transformations ◽  
Arbitrary Vector ◽  
Full Transformation ◽  
Full Transformation Semigroup

In this paper we consider the characterisation of those elements of a transformation semigroup S which are a product of two proper idempotents. We give a characterisation where S is the endomorphism monoid of a strong independence algebra A, and apply this to the cases where A is an arbitrary set and where A is an arbitrary vector space. The results emphasise the analogy between the idempotent generated subsemigroups of the full transformation semigroup of a set and of the semigroup of linear transformations from a vector space to itself.

scholarly journals A note on subgroups of automorphism groups of full shifts

2016 ◽  
Vol 38 (4) ◽  
pp. 1588-1600 ◽  
Author(s):  
VILLE SALO
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Endomorphism Monoid ◽  
Direct Products ◽  
Free Products ◽  
Group Case ◽  
Sofic Shift ◽  
Full Shift

We discuss the set of subgroups of the automorphism group of a full shift and submonoids of its endomorphism monoid. We prove closure under direct products in the monoid case and free products in the group case. We also show that the automorphism group of a full shift embeds in that of an uncountable sofic shift. Some undecidability results are obtained as corollaries.

Absolute graphs with prescribed endomorphism monoid

2007 ◽  
Vol 76 (2) ◽  
pp. 256-267 ◽  
Author(s):  
Manfred Droste ◽  
Rüdiger Göbel ◽  
Sebastian Pokutta
Keyword(s):  
Endomorphism Monoid

Right Invariant Right Hereditary Rings

1974 ◽  
Vol 26 (5) ◽  
pp. 1186-1191 ◽  
Author(s):  
H. H. Brungs
Keyword(s):  
Principal Ideal ◽  
Prime Ideals ◽  
Maximal Orders ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Dedekind Domains ◽  
Left Order ◽  
Principal Ideal Domains

Let R be a right hereditary domain in which all right ideals are two-sided (i.e., R is right invariant). We show that R is the intersection of generalized discrete valuation rings and that every right ideal is the product of prime ideals. This class of rings seems comparable with (and contains) the class of commutative Dedekind domains, but the rings considered here are in general not maximal orders and not Dedekind rings in the terminology of Robson [9]. The left order of a right ideal of such a ring is a ring of the same kind and the class contains right principal ideal domains in which the maximal right ideals are two-sided [6].

Idempotent rank in endomorphism monoids of finite independence algebras

2007 ◽  
Vol 137 (2) ◽  
pp. 303-331 ◽  
Author(s):  
R. Gray
Keyword(s):  
Proper Ideal ◽  
Endomorphism Monoid ◽  
Endomorphism Monoids ◽  
Idempotent Rank

In 1992, Fountain and Lewin showed that any proper ideal of an endomorphism monoid of a finite independence algebra is generated by idempotents. Here the ranks and idempotent ranks of these ideals are determined. In particular, it is shown that when the algebra has dimension greater than or equal to three the idempotent rank equals the rank.

scholarly journals PALINDROMES AND ORDERINGS IN ARTIN GROUPS

2010 ◽  
Vol 19 (02) ◽  
pp. 145-162 ◽  
Author(s):  
FLORIAN DELOUP
Keyword(s):  
Finite Type ◽  
Braid Group ◽  
Type A ◽  
Reverse Order ◽  
Artin Groups ◽  
Left Order ◽  
Group B ◽  
Half Twist

The braid group Bn, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism rev : Bn →Bn, [Formula: see text], defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation τ : x ↦ Δ-1xΔ by the generalized half-twist (Garside element). More generally, the involution rev is defined for all Artin groups (equipped with Artin's presentation) and the involution τ is defined for all Artin groups of finite type. A palindrome is an element invariant under rev. We study palindromes and palindromes invariant under τ in Artin groups of finite type. Our main results are the injectivity of the map [Formula: see text] in all finite-type Artin groups, the existence of a left-order compatible with rev for Artin groups of type A, B, D, and the existence of a decomposition for general palindromes. The uniqueness of the latter decomposition requires that the Artin groups carry a left-order.

The endomorphism monoid of a free trioid of rank 1

2016 ◽  
Vol 76 (3) ◽  
pp. 355-366 ◽  
Author(s):  
Yurii V. Zhuchok
Keyword(s):  
Endomorphism Monoid

scholarly journals IDEMPOTENT RANK IN THE ENDOMORPHISM MONOID OF A NONUNIFORM PARTITION

2015 ◽  
Vol 93 (1) ◽  
pp. 73-91 ◽  
Author(s):  
IGOR DOLINKA ◽  
JAMES EAST ◽  
JAMES D. MITCHELL
Keyword(s):  
Endomorphism Monoid ◽  
Generating Sets ◽  
Idempotent Rank ◽  
Finite Set ◽  
Uniform Case

We calculate the rank and idempotent rank of the semigroup ${\mathcal{E}}(X,{\mathcal{P}})$ generated by the idempotents of the semigroup ${\mathcal{T}}(X,{\mathcal{P}})$ which consists of all transformations of the finite set $X$ preserving a nonuniform partition ${\mathcal{P}}$. We also classify and enumerate the idempotent generating sets of minimal possible size. This extends results of the first two authors in the uniform case.

scholarly journals On the number of graphs with a given endomorphism monoid

2010 ◽  
Vol 310 (3) ◽  
pp. 376-384
Author(s):  
Václav Koubek ◽  
Vojtěch Rödl
Keyword(s):  
Endomorphism Monoid

scholarly journals v*-ALGEBRAS, INDEPENDENCE ALGEBRAS AND LOGIC

2011 ◽  
Vol 21 (07) ◽  
pp. 1237-1257 ◽  
Author(s):  
JOÃO ARAÚJO ◽  
MÁRIO EDMUNDO ◽  
STEVEN GIVANT
Keyword(s):  
Vector Space ◽  
Universal Algebra ◽  
Semigroup Theory ◽  
Endomorphism Monoid ◽  
Transformation Monoid ◽  
Middle Period ◽  
The 1960S

Independence algebras were introduced in the early 1990s by specialists in semigroup theory, as a tool to explain similarities between the transformation monoid on a set and the endomorphism monoid of a vector space. It turned out that these algebras had already been defined and studied in the 1960s, under the name of v*-algebras, by specialists in universal algebra (and statistics). Our goal is to complete this picture by discussing how, during the middle period, independence algebras began to play a very important role in logic.

Sign in / Sign up

Export Citation Format

Share Document