Topic
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.
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TL;DR: In this paper, the sub-power membership problem (SMP) for finite semigroups was shown to be NP-complete for bands, and the greatest variety of bands all of whose finite members induce a tractable SMP was determined.
Abstract: Fix a finite semigroup $S$ and let $a_1,\ldots,a_k, b$ be tuples in a direct power $S^n$. The subpower membership problem (SMP) for $S$ asks whether $b$ can be generated by $a_1,\ldots,a_k$. For bands (idempotent semigroups), we provide a dichotomy result: if a band $S$ belongs to a certain quasivariety, then $SMP(S)$ is in P; otherwise it is NP-complete.
Furthermore we determine the greatest variety of bands all of whose finite members induce a tractable SMP. Finally we present the first example of two finite algebras that generate the same variety and have tractable and NP-complete SMPs, respectively.
5 citations
TL;DR: In this paper, the X-confluence of a quadratic algebra A defined by generators and relations can be interpreted as an equality between two idempotent endomorphisms acting on tensors of degree three (X is the ordered set of generators).
5 citations
TL;DR: In this article , the authors studied the Finite Basis Problem for additively idempotent semirings whose multiplicative reducts are inverse semigroups, and they showed that for any such semiring, there is no finite identity basis.
Abstract: We study the Finite Basis Problem for finite additively idempotent semirings whose multiplicative reducts are inverse semigroups. In particular, we show that each additively idempotent semiring whose multiplicative reduct is a nontrivial rook monoid admits no finite identity basis, and so do almost all additively idempotent semirings whose multiplicative reducts are combinatorial inverse semigroups.
5 citations
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TL;DR: In this article, a decomposition of the unit element of a cellular algebra into orthogonal idempotents (not necessary primitive) satisfying some conditions is considered. And the relation of standard modules, simple modules and decomposition numbers among these algebras is studied.
Abstract: For a cellular algebra $\A$ with a cellular basis $\ZC$, we consider a decomposition of the unit element $1_\A$ into orthogonal idempotents (not necessary primitive) satisfying some conditions. By using this decomposition, the cellular basis $\ZC$ can be partitioned into some pieces with good properties. Then by using a certain map $\a$, we give a coarse partition of $\ZC$ whose refinement is the original partition. We construct a Levi type subalgebra $\aA$ of $\A$ and its quotient algebra $\oA$, and also construct a parabolic type subalgebra $\tA$ of $\A$, which contains $\aA$ with respect to the map $\a$. Then, we study the relation of standard modules, simple modules and decomposition numbers among these algebras. Finally, we study the relationship of blocks among these algebras.
5 citations
5 citations