Ideal Extension Research Articles

Remarks on varieties of involution bands

In the present paper, we study varieties consisting of bands (idempotent semigroups) endowed with an involutorial antiautomorphism * as a fundamental operation. Our principal aim is here to provide an insight to some classes of these varieties from the structural point of view, especially in terms of semilattice decompositions, subdirect products and ideal extensions. In the course of such considerations, we shall extend the result of C. L. Adair [1], who described the lattice of all varieties of bands with a regular involution (i.e. with the identity x = xx * x) We depict a broader lattice of involution band varieties, which incorporates Adair’s lattice.

On Compactness with Respect to Countable Extensions of Ideals and the Generalized Banach Category Theorem

We extend a theorem of Hamlett and Jankovic by proving that if a topological space (X, τ) is compact with respect to the countable extension of I, then the local function A*(I) of every subset A of X with respect to τ and I is a compact subspace with respect to the extension Ĩ in A* (I). We also give a generalized version of the Banach category theorem.

Basic properties of e-inversive semigroups

Generalizing a property of regular resp. finite semigroups a semigroup S is called E-(0-) inversive if for every a ∈ S4(a ≠ 0) there exists x ∈ S such that ax (≠ 0) is an idempotent. Several characterizations are given allowing to identify the (completely, resp. eventually) regular semigroups in this class. The case that for every a ∈ S4(≠ 0) there exist x,y ∈ S such that ax = ya(≠ 0) is an idempotent, is dealt with also. Ideal extensions of E- (0-)inversive semigroups are studied discribing in particular retract extensions of completely simple semigroups. The structure of E- (0-)inversive semigroups satisfying different cancellativity conditions is elucidated. 1991 AMS classification number: 20M10.

Commutative ideal extensions of abelian groups

Ideal extensions of semigroups were introduced by Clifford in [1] and since then they have been widely studied (see for instance [2]). Our aim here is to characterize commutative ideal extensions of Abelian groups. We show that they are those commutative semigroups with an idempotent Archimedean element, or equivalently, those commutative semigroups E such that E/R is an Abelian group, where R is the least congruence that makes E/R cancellative. These characterizations give rise to an algorithm for deciding from a presentation of a finitely generated commutative monoid whether it is an ideal extension of an Abelian group. Finally we present a procedure that enables us to compute the set of idempotents of a finitely generated commutative monoid. All semigroups appearing in this paper are commutative and for this reason in the sequel we sometimes omit the adjective commutative whenever we refer to a commutative semigroup (the same holds for monoids). The authors would like to thank the referee for his/her comments and suggestions.

Separative cancellation for projective modules over exchange rings

A separative ring is one whose finitely generated projective modules satisfy the propertyA⊕A⋟A⊕B⋟B⊕B⇒A⋟B. This condition is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective modules over exchange rings. It is shown that the class of separative exchange rings is very broad, and, notably, closed under extensions of ideals by factor rings. That is, if an exchange ringR has an idealI withI andR/I both separative, thenR is separative.

On the ideal extensions in ordered semigroups

On the ideal extensions in ordered semigroups

Orthogonal sums of 0-σ-simple semigroups

Throughout this paper, Z+ will denote the set of all positive integers and S = S0 means that S is a semigroup with the zero 0. If S = S0, we will write 0 instead {0} and if A is a subset of S, we will write A• = A−0, A0 = A∪0, A′ = (S − A)0. For an element a of a semigroup S, J(a) will denote the principal ideal of S generated by a. Let S = S0. An element a of S is a nilpotent if there exists n ∈ Z+ such that an = 0. The set of all nilpotents of S is denoted by Nil(S). If S = Nil(S), then S is a nil-semigroup, otherwise it is non-nil . If S2 = 0, then S is a null-semigroup, otherwise it is non-null . An ideal I of S is a nil-ideal of S if I is a nil-semigroup. An ideal extension S of a semigroup K is a nil-extension of K if S/K is a nil-semigroup. A semigroup S is intra-π-regular if for every a ∈ S there exists n ∈ Z+ such that an ∈ Sa2nS. A semigroup S is completely π-regular if for every a ∈ S there exists n ∈ Z+ and x ∈ S such that an = anxan, anx = xan. An ideal A of a semigroup S is prime if for all a, b ∈ S, aSb ⊆ A implies that either a ∈ A or b ∈ A, or, equivalently, if for ideals M, N of S, MN ⊆ A implies that either M ⊆ A or N ⊆ A. An ideal A of a semigroup S is semiprime if for every a ∈ S, aSa ⊆ A implies a ∈ A. An ideal A of a semigroup S is completely semiprime if for a ∈ S, a2 ∈ A implies a ∈ A. A subset A of a semigroup S is consistent if for x, y ∈ S, xy ∈ A implies x, y ∈ A. A subset A of a semigroup S = S0 is 0-consistent if A• is consistent. Let S be a semigroup. If a, b ∈ S, then a |b if b ∈ J(a), and a−→b if a | bn, for some n ∈ Z+. By−→n, n ∈ Z+, and −→∞ we will denote the n-th power and the transitive closure of the relation−→, respectively. For a ∈ S, n ∈ Z+,

On subtractive varieties II: General properties

As a sequel to [23] we investigate ideal properties focusing on subtractive varieties. After having listed a few basic results, we give several characterizations of the commutator of ideals and prove, for example, that it commutes with finite direct products. We deal with the ideal extension property (IEP) and with related commutator properties, showing for instance that IEP implies that the commutator commutes with restriction to subalgebras. Then we characterize varieties with distributive ideal lattices and relate this property with (a form of) equationally definable principal ideals and with IEP. Then, at the other extreme, we deal with Abelian and Hamiltonian properties (of ideals and congruences), giving for example a purely ideal theoretic characterization of varieties of Abelian groups with linear operations. To finish with, we present a few examples aiming at vindicating our work.

Ideal structure for superconformal semigroups

The role of univertible transformations in superstring theories is discussed. A new parametrization of superconformal groups is presented that permits their ideal extensions—superconformal semigroups—to be constructed adequately. These semigroups consist of a group part containing the standard superconformal transformations and an ideal part whose abstract structure is analyzed in detail. A classification of all elements in terms of different indices of nilpotency is performed. An ideal series is constructed, the generalized Green's “vector” and “tensor” relations are defined, and some types of quasi-characters separating the elements of superconformal semigroups are introduced. The need for similar constructions in other supersymmetric and superquantum models is pointed out.

Passive compliance and active force generation in the guinea pig outer hair cell.

1. Cochlear outer hair cells 20-80 microns in length were compressed axially in vitro using calibrated glass fibers mounted on a piezoelectric actuator. 2. When driven by rectangular pulses in the compression direction, the motion of the fiber tip consisted of a rapid initial compression that was complete in 10-20 ms followed by a smaller compression of slower time course. 3. The initial fiber deflections were found to be linear in amplitude for compressions up to 400 nm. The axial compliances of outer hair cells were calculated from the difference between the fiber tip motions when unattached and when in contact with a cell. Axial compliances were found to be in the range of 0.04-1.2 km/N for 149 cells. The axial compliance was an increasing function of cell length. 4. The peak forces generated by electrically stimulated outer hair cells were measured from the deflection of a glass fiber when the cells were stimulated by sinusoidal voltage commands. The slope gains of force generation (force generated per mV of command at the cell membrane) were estimated to range from 0.01 to 100 pN/mV. Most of the results fell in the range of 0.1-20 pN/mV. 5. When the apparent stiffness of the fiber was increased by moving the cell closer to the fiber base, the peak amplitude of the fiber deflection generated by the cell decreased and the peak force increased, for the same sinusoidal voltage command. 6. The results of the previous experiment were interpreted in the light of a model of outer hair cell motility in which an ideal extension generating element is in series with an internal stiffness element. This internal stiffness was then calculated for 13 cells. 7. The internal stiffnesses of cells calculated by the above procedure were found to be positively correlated with the axial stiffness measurements obtained for the same cells. 8. The implications of the above results for the effectiveness of outer hair cell motility in vivo are discussed.

Factorization of Prime Ideal Extensions in Dedekind Domains

In this paper, improving on the results of Del Corso (1992), we describe a method to factorize prime ideal extensions in Dedekind domains. This method needs factorization modulo P of a polynomial in at least two variables.

Ideal extensions of rings-some topological aspect

Ideal extensions of rings-some topological aspect

The kernel and trace operators for ideal extensions of regular semigroups

AbstractLet V be a regular semigroup and an ideal extension of a semigroup S by a semigroup Q Congruences on V can be represented by triples of the form (σ, P, τ), here called admissible, where a is a congruence on S, P is an ideal of Q and τ is a O-restricted congruence on Q/P satisfying certain conditions. We characterize the trace relation T on V in terms of admissible triples. When the extension V of S is strict, for a congruence v on V given in terms of an admissible triple, we characterize vK, vK, vT and vT again in terms of admissible triples.

Response surface methodology for non-equispaced experimental configurations applied to a discrete and stochastic simulator of a manufacturing company

This work is based on the construction and use of a SLAM II PC-based simulator for a medium-sized company manufacturing machine tools designed to work glass. Response surface methods and several experimental design techniques were applied to this company to optimize simulator performances. Since there is a feedback loop on the actual manufacturing system, the theory of orthogonal polynomials was applied to the case-study relative to a planned experimental configuration of non-equispaced tests performed on the real system, which were based on analysis characteristics. This discussion is the ideal extension of the basic case involving equispaced configurations. A summary is presented of the results obtained from the application of a 3X4 mixed level factorial experiment implemented with a discrete and stochastic industrial simulator.

Ideal extensions of Γ-rings

AbstractGiven Γ-rings N1 and N2, a construction similar to the Everett sum of rings to find all possible extensions of N1 by N2 is given. Unlike the case of rings, it is not possible to find for any Γ-ring M an ideal extension that has a unity. Furthermore, contrary to the ring case, a Γ-ring with unity can not be characterized as a Γ-ring which is a direct summand in every extension thereof.

The congruence lattice of an ideal extension of semigroups

Let S be an ideal of a semigroup V. In such a case, V is an (ideal) extension of S by T = V/S. The problem considered in [2] is the construction of all congruences on V in terms of congruences on S and T. This did not succeed for all congruences but it did for those congruences whose restriction to S is weakly reductive. If the extension is strict, more precise constructions are also given there. With some relatively weak restrictions on S, we are able to obtain in this way all congruences on V in the form indicated above.

Factorization of prime ideal extensions in number rings

Following an idea of Kronecker, we describe a method for factoring prime ideal extensions in number rings. The method needs factorization of polynomials in many variables over finite fields, but it works for any prime and any number field extension.

Factorization of Prime Ideal Extensions in Number Rings

Factorization of Prime Ideal Extensions in Number Rings

Extensions of fuzzy subrings and fuzzy ideals

In this paper we give necessary and sufficient conditions for a fuzzy subring or a fuzzy ideal A of a commutative ring R to be extended to one A e of a commutative ring S containing R as a subring such that A and A e have the same image. One of the applications of these results gives a criterion for a fuzzy subring of an integral domain R to be extendable to a fuzzy subfield of the quotient field of R.

Properties of operators of extension and contraction of ideals of algebras of invariants of reductive groups in positive characteristic

Properties of operators of extension and contraction of ideals of algebras of invariants of reductive groups in positive characteristic