Idempotence

About: Idempotence is a research topic. Over the lifetime, 1860 publications have been published within this topic receiving 19976 citations. The topic is also known as: idempotent.

Banach algebras generated by two idempotents and one flip

Abstract: The paper is concerned with the invertibility of elements in a Banach algebra generated by two idempotents and one flip element. A symbol in form of a 2 × 2 matrix function which is defined on some Hausdorff compact will be constructed and a complete description of will be given.

The Thick Subcategory Generated by the Trivial Module

Abstract: This paper is a report of my lecture at the International Conference on the Representation Theory of Algebras which preceded the Euroconference on Infinite Length Modules. While it is true that the aim of the research is the homological algebra of finitely presented modules, a major point of the lecture was the role that infinitely generated modules could play in the investigation. The methods of idempotent modules and the theory of support varieties for infinite dimensional modules have already had a significant impact on group representation theory. It seems certain that there will be a lot more to follow. I am honored by the invitation to include the report in the conference proceedings, and I would like to thank the organizers of the conference and the workshop for the stimulating experience.

Idempotent mathematics and mathematical physics : international workshop, February 3-10, 2003, Erwin Schrödinger International Institute for Mathematical Physics, Vienna, Austria

Abstract: The Maslov's dequantization, idempotent and tropical mathematics: A very brief introduction by G. L. Litvinov Set coverings and invertibility of functional Galois connections by M. Akian, S. Gaubert, and V. Kolokoltsov Discrete max-plus spectral theory by M. Akian, S. Gaubert, and C. Walsh Dequantization of coadjoint orbits: Moment sets and characteristic varieties by A. Baklouti On the combinatorial aspects of max-algebra by P. Butkovic Max-plus convex sets and functions by G. Cohen, S. Gaubert, J.-P. Quadrat, and I. Singer Algebras of Lukasiewicz's logic and their semiring reducts by A. Di Nola and B. Gerla Max-plus approaches to continuous space control and dynamic programming by W. H. Fleming and W. M. McEneaney A blow-up phenomenon in the Hamilton-Jacobi equation in an unbounded domain by K. Khanin, D. Khmelev, and A. Sobolevskii The dequantization transform and generalized Newton polytopes by G. L. Litvinov and G. B. Shpiz An object-oriented approach to idempotent analysis: Integral equations as optimal control problems by P. Loreti and M. Pedicini Traffic assignment & Gibbs-Maslov semirings by P. Lotito, J.-P. Quadrat, and E. Mancinelli Viscosity solutions on Lagrangian manifolds and connections with tunnelling operators by D. McCaffrey Applications of the generated pseudo-analysis to nonlinear partial differential equations by E. Pap A generalization of the utility theory using a hybrid idempotent-probabilistic measure by E. Pap Amoebas: Their spines and their contours by M. Passare and A. Tsikh First steps in tropical geometry by J. Richter-Gebert, B. Sturmfels, and T. Theobald On minimax and idempotent generalized weak solutions to the Hamilton-Jacobi equation by I. V. Roublev Dequantisation: Semi-direct sums of idempotent semimodules by E. Wagneur On (min,max,+)-inequalities by J. van der Woude and G. J. Olsder Solution of some max-separable optimization problems with inequality constraints by K. Zimmermann.

Inertial coefficient rings and the idempotent lifting property

Abstract: A commutative ring R with identity is called an inertia! coefficient ring if every finitely generated ^-algebra A with A/N separable over R contains a separable ^-subalgebra S of A such that A = S + N, where N is the Jacobson radical of A. We say A has the idempotent lifting property if every idempotent in A/N is the image of an idempotent in A. Our main theorem is that any finitely generated algebra over an inertial coefficient ring has the idempotent lifting property. All rings contain an identity; all subrings contain the identity of the overring; all homomorphisms preserve the identity. Throughout R denotes a commutative ring and A an R -algebra which is finitely generated as an fi-module. The Jacobson radical of a ring B is denoted rad(fi) and throughout rad(A) = A. A separable fi-subalgebra 5 of A such that A = S + A is called an inertial subalgebra. If every finitely generated fi-algebra A with A/N fi-separable has an inertial subalgebra, R is called an inertial coefficient ring. The basic properties of inertial subalgebras and inertial coefficient rings can be found in [7]. If / is an ideal of a ring B we call (B, I) an L. I. pair (lifting idempotent pair) if every idempotent in the factor ring fi// is the image of an idempotent in B; if (A, A) is an L. I. pair we say A has the idempotent lifting property. Our main theorem is motivated by a conjecture of E. C. Ingraham that if every finitely generated fi-algebra has the idempotent lifting property then R is an inertial coefficient ring. Our proof of the converse of this conjecture has as a corollary that an inertial coefficient ring is a Hensel ring (see [5], [6], and [10] for definition and properties of Hensel rings), recalling the role Hensel local rings have played in generalizations of the Wedderburn Principal Theorem by Azumaya, Ingraham, and W. C. Brown. A second immediate consequence of our main theorem is that when R is an inertial coefficient ring, two inertial subalgebras of an fi-algebra A are conjugate under an inner automorphism of A, generalizing Malcev's uniqueness statement to Wedderburn's Principal Theorem. Received by the editors April 14, 1976. AMS (MOS) subject classifications (1970). Primary 16A16, 16A32; Secondary 13J15.

Three Classes of Closed Sets of Monomials

Abstract: We consider three classes of monomials: unary, binary with at least one linear literal, and idempotent binary. A functionally closed set containing a unary monomial may or may not contain identity, and it can be generated by a singleton or by an arbitrary set of monomials. This induces four different classes of functionally closed sets of unary monomials. These classes are ordered by set inclusion and the emphasis is put on minimal, maximal, least and greatest elements. For binary monomials with at least one linear literal, we describe the structure of the set of clones generated by singletons. Finally, for idempotent binary monomials, we determine the least and the greatest element.