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Rational semimodules over the max-plus semiring and geometric approach to discrete event systems

Stéphane Gaubert, Ricardo Katz (2004)

Kybernetika

We introduce rational semimodules over semirings whose addition is idempotent, like the max-plus semiring, in order to extend the geometric approach of linear control to discrete event systems. We say that a subsemimodule of the free semimodule 𝒮 n over a semiring 𝒮 is rational if it has a generating family that is a rational subset of 𝒮 n , 𝒮 n being thought of as a monoid under the entrywise product. We show that for various semirings of max-plus type whose elements are integers, rational semimodules...

Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices

Gábor Czédli (2024)

Mathematica Bohemica

Following G. Grätzer and E. Knapp (2007), a slim planar semimodular lattice, SPS lattice for short, is a finite planar semimodular lattice having no M 3 as a sublattice. An SPS lattice is a slim rectangular lattice if it has exactly two doubly irreducible elements and these two elements are complements of each other. A finite poset P is said to be JConSPS-representable if there is an SPS lattice L such that P is isomorphic to the poset J ( Con L ) of join-irreducible congruences of L . We prove that if 1 < n ∈ ℕ and...

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