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.

Third-power associative absolute valued algebras with a nonzero idempotent commuting with all idempotents

Abstract: This paper deals with the determination of the absolute valued algebras with a nonzero idempotent commuting with the remaining idempotents and satisfying x2x = xx2 for every x. We prove that, in addition to the absolute valued algebras R, C, H, or O of the reals, complexes, division real quaternions or division real octonions, one such absolute valued algebra A can also be isometrically isomorphic to some of the absolute valued algebras C. H or O, obtained from C, H, and O by imposing a new product defined by multiplying the conjugates of the elements. In particular, every absolute valued algebra having the above properties is finite-dimensional. This generalizes some well known theorems of Albert, Urbanik and Wright, and El-Mallah.

SO.3/ homology of graphs and links

Abstract: Mikhail Khovanov [11] introduced a categorification of the Temperley–Lieb algebra. Recently, the first two authors [4] showed that there are chain complexes within this construction that become the Jones–Wenzl projectors in the image of the Grothendieck group K0 . These chain complexes are unique up to homotopy and idempotent with respect to the tensor product: C C ' C . It is now well-known (see Fendley and Krushkal [5]) that the chromatic algebra and the SO.3/ Birman–Murakami–Wenzl algebra can be constructed using the second Jones–Wenzl projector. In this paper we use the formulation of Bar-Natan [1] to extend the original categorification of the Temperley–Lieb algebra to categorifications of the SO.3/ BMW algebra and the chromatic algebra. Previous work of Helme-Guizon and Rong [9] and Stosic [15] on the categorification of the chromatic polynomial has been focused on constructions which are in many respects independent of structural choices such as the Frobenius algebra. In this paper we obtain an essentially unique categorification of the chromatic polynomial of planar graphs.

Using Last Fixed Points to Characterize Formal Computations of Non-Deterministic Equations

Abstract: We extend the least fixed point techniques to the case of non-deterministic equations. Commutative idempotent semigroups in the category of ω-complete posets are used as non-deterministic domains.

Essential operations in centralizer clones

Abstract: For an algebra A and for any finite n ≥ 2, a homomorphism \(h:A^{n} \rightarrow A\) is essentially n-ary if it depends on all its variables. The existence of such an h brings the integer n into the set S(cA) of all significant arities of the centralizer clone cA of the algebra A. For idempotent algebras, algebras with zero and congruence distributive algebras, the set S(cA) must be an order ideal in ω = {0, 1, . . . } or in ω \ {0} or in ω \ {0, 1}, and we construct such algebras. On the other hand, there exist algebras with two unary operations whose centralizer clones have essential arities that form an order filter of ω. We also construct (0, 1)-lattices for which S(cA) is a prescribed finite order ideal of ω \ {0} or of ω \ {0, 1}, and whose endomorphism monoid End01 A of all its (0, 1)-endomorphisms is isomorphic to a prescribed one. Finally, for a product K × L of (0, 1)-lattices (and other algebras), we give conditions under which the set S(c(K × L)) is the smallest possible, and show why these conditions are needed. In the process we prove that for any lattice A, the category \({\mathbb{L}}[A]\) of all lattices containing A and of their lattice homomorphisms is almost universal.