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- Title
Proper/Residually-Finite Idempotent Semirings.
- Authors
Griffing, Gary
- Abstract
An idempotent semiring (= ISR) is called L-E if its underlying additive Abelian semigroup is generated by join-primes. Not all ISRs are L-E; not even when finite. The submodule of 'linear-recognizable' elements of an ISR $\mathcal {M}$ is denoted $\mathcal {C}\mathcal {M}$, and $\mathcal {M}$ is called proper if there are enough elements in $\mathcal {C}\mathcal {M}$ to separate points. If there are enough finite-index congruences to separate points, $\mathcal {M}$ is called residually-finite. Finite and proper ISRs are always residually-finite, but finite ISRs are not always proper, unless they are L-E. For certain classes of ISRs, conditions are given to guarantee proper and residual-finiteness. Among these is one which requires that the compact elements of the linear dual of $\mathcal {M}$ belong to $\mathcal {C}\mathcal {M}$. Another condition requires that the recognizable subsets of a certain underlying monoid remain recognizable under the closure operator relative to a certain natural topology. These conditions are automatic for any finite L-E ISR, or any L-E ISR arising from a bounded, distributive lattice. Thus, a large class of proper/residually-finite ISRs exists. Moreover, the theorem of Malcev for semigroups (finitely-generated, commutative implies residually-finite) is shown to fail for ISRs in general.
- Subjects
IDEMPOTENTS; SEMIRINGS (Mathematics); ABELIAN semigroups; SEMILATTICES; MATHEMATICAL bounds; TOPOLOGY; MONOIDS
- Publication
Semigroup Forum, 2013, Vol 86, Issue 3, p486
- ISSN
0037-1912
- Publication type
Article
- DOI
10.1007/s00233-013-9484-9