Ideal Extension Research Articles

Semilattice composition of ordered semigroups

We study the semilattice composition of ordered semigroups (a concept opposite to that of the semilattice decomposition), using the ideal extensions.

Ideal extensions of graph algebras

Let A and B be graph algebras. In this paper we present the notion of an ideal in a graph algebra and prove that an ideal extension of A by B always exists. We describe (up to isomorphism) all such extensions.

Lie algebras and algebraic groups

Preface 1. Results on topological spaces 1.1 Irreducible sets and spaces 1.2 Dimension 1.3 Noetherian spaces 1.4 Constructible sets 1.5 Gluing topological spaces 2. Rings and modules 2.1 Ideals 2.2 Prime and maximal ideals 2.3 Rings of fractions and localization 2.4 Localization of modules 2.5 Radical of an ideal 2.6 Local rings 2.7 Noetherian rings and modules 2.8 Derivations 2.9 Module of differentials 3. Integral extensions 3.1 Integral dependence 3.2 Integrally closed rings 3.3 Extensions of prime ideals 4. Factorial rings 4.1 Generalities 4.2 Unique factorization 4.3 Principal ideal domains and Euclidean domains 4.4 Polynomial and factorial rings 4.5 Symmetric polynomials 4.6 Resultant and discriminant 5. Field extensions 5.1 Extensions 5.2 Algebraic and transcendental elements 5.3 Algebraic extensions 5.4 Transcendence basis 5.5 Norm and trace 5.6 Theorem of the primitive element 5.7 Going Down Theorem 5.8 Fields and derivations 5.9 Conductor 6. Finitely generated algebras 6.1 Dimension 6.2 Noether's Normalization Theorem 6.3 Krull's Principal Ideal Theorem 6.4 Maximal ideals 6.5 Zariski topology 7. Gradings and filtrations 7.1 Graded rings and graded modules 7.2 Graded submodules 7.3 Applications 7.4 Filtrations 7.5 Grading associated to a filtration 8. Inductive limits 8.1 Generalities 8.2 Inductive systems of maps 8.3 Inductive systems of magmas, groups and rings 8.4 An example 8.5 Inductive systems of algebras 9. Sheaves of functions 9.1 Sheaves 9.2 Morphisms 9.3 Sheaf associated to a presheaf 9.4 Gluing 9.5 Ringed space 10. Jordan decomposition and some basic results on groups 10.1 Jordan decomposition 10.2 Generalities on groups 10.3 Commutators 10.4 Solvable groups 10.5 Nilpotent groups 10.6 Group actions 10.7 Generalities on representations 10.8 Examples 11. Algebraic sets 11.1 Affine algebraic sets 11.2 Zariski topology 11.3 Regular functions 11.4 Morphisms 11.5 Examples of morphisms 11.6 Abstract algebraic sets 11.7 Principal open subsets 11.8 Products of algebraic sets 12. Prevarieties and varieties 12.1 Structure sheaf 12.2 Algebraic prevarieties 12.3 Morphisms of prevarieties 12.4 Products of prevarieties 12.5 Algebraic varieties 12.6 Gluing 12.7 Rational functions 12.8 Local rings of a variety 13. Projective varieties 13.1 Projective spaces 13.2 Projective spaces and varieties 13.3 Cones and projective varieties 13.4 Complete varieties 13.5 Products 13.6 Grassmannian variety 14. Dimension 14.1 Dimension of varieties 14.2 Dimension and the number of equations 14.3 System of parameters 14.4 Counterexamples 15. Morphisms

Maximal Dense Ideal Extensions of Locally Inverse Semigroups

Every locally inverse semigroup has a maximal dense ideal extension within the class of all locally inverse semigroups.

Unit and kernel systems in algebraic frames

One considers the poset of dense, coherent frame quotients of an algebraic frame with the finite intersection property, which are compact. It is shown that there is a smallest such, the frame of d-elements. However, unless the frame is already compact there is no largest such quotient. With the additional assumption of disjointification on the frame, one then studies the maximal ideal spaces of these quotients and the relationship to covers of compact spaces. Several applications are considered, with considerable attention to the frame quotients defined by extension of ideals of a commutative ring A to a ring extension; this type of frame quotient is considered both with and without an underlying lattice structure on the rings.

Homoflatness on Ideal Extensions

We consider monoids $S=G\;\dot{\cup}\; I$ where $G$ is a group and $I$ is an ideal of $S$ and show that if an $S$-act is principally weakly homoflat or weakly homoflat as an $I^1$-act, then it has these properties as an $S$-act. We also show that an $S$-act which is (weakly) pullback flat, equalizer flat, (principally) weakly kernel flat, translation kernel flat or satisfies Condition $(E’)$ as an $I^1$-act may not generally have these properties as an $S$-act. The flatness notions considered in this paper were introduced in {\it V. Laan, Pullbacks and flatness properties of acts I, Comm. Alg. ${\bf 29}(2)$ (2001), 829--850}.

EXTENSIONS OF CLIFFORD SUBSEMIGROUPS OF THE FINITE SYMMETRIC INVERSE SEMIGROUP#

ABSTRACT We describe maximal Clifford subsemigroups of the finite symmetric inverse semigroup I n of the set X n − { 1, 2,…,n } and obtain their complete classification, determine the number and the order of such subsemigroups, give a criterion for two maximal Clifford subsemigroups of I n to be isomorphic, and determine (up to isomorphism) the number of such subsemigroups. Further, both maximal ideal extensions and maximal nil extensions of any Clifford subsemigroup S of I n are described. In particular, we show that S has a unique maximal ideal extension in I n and, up to isomorphism, S has a unique maximal nil extension in I n .

The Ideal Extension and Congruence Extension Properties for Compact Semigroups

The aim of this paper is to study the congruence extension property and the ideal extension property for compact semigroups. We present a characterization of compact semigroups with the ideal extension property and prove that each compact semigroup with the congruence extension property also has the ideal extension property.

Ideal extensions of ordered sets

The ideal extensions of semigroups—without order—have been first considered by Clifford (1950). In this paper, we give the main theorem of the ideal extensions for ordered sets. If P, Q are disjoint ordered sets, we construct (all) the ordered sets V which have an ideal P′ which is isomorphic to P, and the complement of P′ in V is isomorphic to Q. Conversely, we prove that every extension of an ordered set P by an ordered set Q can be so constructed. Illustrative examples of the main theorem in case of finite ordered sets are given.

Extending conditional likelihood in models for stratified binary data

The conditional likelihood is widely used in logistic regression models with stratified binary data. In particular, it leads to accurate inference for the parameters of interest, which are common to all strata, eliminating stratum-specific nuisance parameters. The modified profile likelihood is an accurate approximation to the conditional likelihood, but has the advantage of being available for general parametric models. Here, we propose the modified profile likelihood as an ideal extension of the conditional likelihood in generalized linear models for binary data, with generic link function. An important feature is that for the implementation we only need standard outputs of routines for generalized linear models. The accuracy of the method is supported by theoretical properties and is confirmed by simulation results.

FINITELY GENERATED COMMUTATIVE ARCHIMEDEAN SEMIGROUPS

In this work commutative Archimedean finitely generated semigroups are characterized in terms of ideal extensions.

Semigroups with the ideal extension property

The aim of this paper is to study semigroups with the ideal extension property. We establish a structure theorem for semigroups with the ideal extension property.

The ideal extension property in compact semigroups

The aim of this paper is to study and characterize compact semigroups with the ideal extension property. We establish a characterization of compact semigroups having the ideal extension property. In particular, we completely determine the structure of such semigroups with the property that regular elements form a subsemigroup, and also the structure of such semigroups with precisely one regular D-class.

Ideal Extensions of Ordered Semigroups†

The ideal extensions of semigroups -without order- have been first considered by Clifford (Clifford, A. H. (1950). Extension of semigroups. Trans. Amer. Math. Soc. 68: 165–173). The aim of this paper is to give the main theorem of the ideal extensions for ordered semigroups.

Extension of Finite Rank Operators and Operator Ideals with the Property (I)

We present criteria and related techniques which help to decide whether the adjoint operator ideal (𝔄*, A*) of an injective and totally accessible maximal Banach ideal (𝔄, A) is itself also totally accessible. This approach (which involves the transfer of the principle of local reflexivity to operator ideals) is based on the extension of finite rank operators, viewed as elements of the adjoint ideal (𝔄*, A*). Using the local properties (I) and (S) of the corresponding product ideal 𝔄* ∘ 𝔏∞, these methods even enable us to show that 𝔏∞ and 𝔏1 cannot be totally accessible – answering an open question of Defant and Floret.

Extensions and submonoids of automatic monoids

In this paper, submonoids and extensions of automatic and p-automatic monoids are studied. The concept of a p-automatic monoid is a variant on the usual concept of an automatic monoid designed to allow a geometric characterization analogous to the group case. In the case of right cancellative monoids, the two concepts coincide. Here, we study rational submonoids of (p-)automatic monoids, being able to show in many cases that (p-)automaticity is inherited. Our sharpest results concern rational subgroups. Also, closure properties are established for various notions of extensions of (p-)automatic monoids, including different types of products, ideal extensions, and Rees matrix constructions.

Commutative ideal extensions of Abelian groups

Commutative ideal extensions of Abelian groups

The Ossy Splint - A New Design of Dynamic Extension Splint

The ideal dynamic extension splint should produce a constant and consistent force through its therapeutic range at 90° to the digit and allow ease of use to aid compliance. Currently available splints work either by stretching a spring or an elastic band via a fixed outrigger pulley.

Finite semigroups with slowly growing p n -sequences

The p n -sequence of a semigroup S is said to be polynomially bounded, if there exist a positive constant c and a positive integer r such that the inequality p n (S) ≤cn r holds for all n≥ 1. In this paper, we fully describe all finite semigroups having polynomially bounded p n -sequences. First we give a characterization in terms of identities satisfied by these semigroups. In the sequel, this result will allow an insight into the structure of such semigroups. We are going to deal with certain ideals and the construction of ideal extension of semigroups. In addition, we supply an effective procedure for deciding whether a finite semigroup has polynomially bounded p n -sequence and give some examples.

Morphometry of late Pleistocene‐Holocene faulting and volcanotectonic relationship in the southern Andes of Colombia

Diffuse seismicity and the active Galeras Volcano characterize the southern Andes of Colombia, whose inhabitants are prone to a high natural risk. We used field geological, structural, and quantitative morphological data to define the geometry and kinematics of late Quaternary faults and the volcanotectonic relationships in this area. The main NE‐SW Buesaco Fault has a late Pleistocene‐Holocene 29‐km rupture length. Offset of dated landforms yields 188 ± 13 m of horizontal, right‐lateral, long‐term cumulative component of slip, the vertical component of uplift of the NW block being 67 ± 12 m. The long‐term net slip rate is 1.48 ± 0.12 mm/yr. The other main structure, here named Aranda Fault, also strikes NE‐SW and is 13 km long. Offset of landforms yields 160 ± 10 m of horizontal, right‐lateral, long‐term component of slip and 3.9 ± 1.9 m of vertical component of uplift of the NW block. The long‐term net slip rate is 1.19 ± 0.08 mm/yr. On the Galeras edifice E‐W to NW‐SE faults have normal motions and slip rates from 0.02 to 0.23 mm/yr. Both the Galeras and a late Pleistocene pyroclastic cone lie along the ideal extension of the Buesaco Fault. Magma uprising in the crust can occur along the NE striking fractures when magma pressure (σmp) is greater than the normal‐to‐fault component of tectonic σ1. Magma uprising in the uppermost crust (i.e., in the cone) can occur by rock cracking along E‐W planes with σmp > σ3 up to NW‐SE planes with σmp greater than shear stress component of the horizontal greatest principal stress (σHmax).