Proper/Residually-Finite Idempotent Semirings

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.