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.

Combinatorial techniques for determining rank and idempotent rank of certain finite semigroups

Abstract: Let τ be a partition of the positive integer n. A partition of the set {1, 2 ,...,n } is said to be of type τ if the sizes of its classes form the partition τ of n. It is known that the semigroup S(τ ), generated by all the transformations with kernels of type τ , is idempotent generated. When τ has a unique non-singleton class of size d, the difficult Middle Levels Conjecture of combinatorics obstructs the application of known techniques for determining the rank and idempotent rank of S(τ ). We further develop existing techniques, associating with a subset U of the set of all idempotents of S(τ ) with kernels of type τ a directed graph D(U ); the directed graph D(U ) is strongly connected if and only if U is a generating set for S(τ ), a result which leads to a proof if the fact that the rank and the idempotent rank of S(τ ) are both equal to max n , n d +1 .

An algebraic approach to multidimensional minimax location problems with Chebyshev distance

Abstract: Minimax single facility location problems in multidimensional space with Chebyshev distance are examined within the framework of idempotent algebra. The aim of the study is twofold: first, to give a new algebraic solution to the location problems, and second, to extend the area of application of idempotent algebra. A new algebraic approach based on investigation of extremal properties of eigenvalues for irreducible matrices is developed to solve multidimensional problems that involve minimization of functionals defined on idempotent vector semimodules. Furthermore, an unconstrained location problem is considered and then represented in the idempotent algebra settings. A new algebraic solution is given that reduces the problem to evaluation of the eigenvalue and eigenvectors of an appropriate matrix. Finally, the solution is extended to solve a constrained location problem.

Weak Multiplier Hopf Algebras II. The source and target algebras

Abstract: In this paper, we continue the study of weak multiplier Hopf algebras. We recall the notions of the source and target maps $\varepsilon_s$ and $\varepsilon_t$, as well as of the source and target algebras. Then we investigate these objects further. Among other things, we show that the canonical idempotent $E$ (which is eventually $\Delta(1)$) belongs to the multiplier algebra $M(B\otimes C)$ where $B=\varepsilon_s(A)$ and $C=\varepsilon_t(A)$ and that it is a separability idempotent. We also consider special cases and examples in this paper. In particular, we see how for any weak multiplier Hopf algebra, it is possible to make $C\otimes B$ (with $B$ and $C$ as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the 'Hopf algebra part' of the original weak multiplier Hopf algebra and only remembers the source and target algebras. It is in turn generalized to the case of any pair of non-degenerate algebras $B$ and $C$ with a separability idempotent $E\in M(B\otimes C)$. We get another example using this theory associated to any discrete quantum group (a multiplier Hopf algebra of discrete type with a normalized cointegral). Finally we also consider the well-known 'quantization' of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras).

Koszul Gradings on Brauer Algebras

Abstract: We show that the Brauer algebra Brd(δ) over the complex numbers for an integral parameter δ can be equipped with a grading. In case δ ≠ 0 it becomes a graded quasi-hereditary algebra which is moreover Morita equivalent to a Koszul algebra. These results are obtained by realizing the Brauer algebra as an idempotent truncation of a certain level two VW-algebra ⩔ cycl d (N) for some large positive integral parameter N . The parameter δ appears here in the choice of a cyclotomic quotient. This cyclotomic VW-algebra arises naturally as an endomorphism algebra of a certain projective module in parabolic category O of type D. In particular, the graded decomposition numbers are given by the associated parabolic Kazhdan-Lusztig polynomials.