Algebraic Structures in Tropical Matrix Semigroups
Summary
Tropical algebra studies structures over the tropical semiring, where addition is defined as taking a minimum (or maximum) and multiplication as ordinary addition. Within this framework, tropical matrix semigroups comprise sets of matrices equipped with tropical multiplication, yielding rich algebraic and combinatorial phenomena. These semigroups serve as natural models for shortest-path problems, network flows and discrete event systems, and connect deeply with tropical geometry, polyhedral combinatorics and automata theory. Key themes include the classification of identities and varieties, faithful matrix representations, the structure of Green’s relations, and the geometry of semigroup actions on polyhedral fans. Recent work has uncovered finite descriptions of these semigroups via generators and relations, elucidated the role of upper triangular and unitriangular matrices over the tropical semiring, and demonstrated how tropical semigroup techniques can solve problems in optimisation, language theory and combinatorial representation theory.
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Algebraic Structures in Tropical Matrix Semigroups publication trend
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Technical terms
Semiring: An algebraic structure with two operations—addition and multiplication—where addition is commutative and associative with an identity, multiplication is associative with an identity, and multiplication distributes over addition.
Tropical semiring: A semiring whose “addition” is given by taking a minimum (or maximum) of two elements and whose “multiplication” is conventional addition.
Matrix semigroup: A set of matrices closed under an associative multiplication operation.
Polyhedral fan: A finite collection of polyhedral cones in a vector space, closed under taking faces and intersections, often encoding combinatorial data of semigroup actions.
Unitriangular matrix: A triangular matrix with all diagonal entries equal to the multiplicative identity; in the tropical setting this often simplifies identity and basis questions.
Variety (in semigroup theory): A class of semigroups defined by a set of identities, closed under taking homomorphic images, subsemigroups and direct products.
References
- Representations and identities of plactic-like monoids. Journal of Algebra (2022).
- Lossy Gossip and Composition of Metrics. Discrete & Computational Geometry (2015).
- Tropical representations and identities of the stylic monoid. Semigroup Forum (2022).
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