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We study conditions under which the lattice $${{\mathrm{\mathbf {Id}}}}\mathbf R$$ Id R of ideals of a given a commutative semiring $${\mathbf {R}}$$ R is complemented. At first we check when the annihilator $$I^*$$ I ∗ of a given ideal I of $${\mathbf {R}}$$ R is a complement of I. Further, we study complements of annihilator ideals. Next we investigate so-called...
Every residuated lattice can be considered as an idempotent semiring. Conversely, if an idempotent semiring is finite, then it can be organized into a residuated lattice. Unfortunately, this does not hold in general. We show that if an idempotent semiring is equipped with an involution which satisfies certain conditions, then it can be organized into a residuated lattice satisfying the double negation...
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